The effect of collisions on the motion of magnetized ions in sheath and presheath plasma regions was investigated through the measurement of ion incident angle of a hydrogen ion at a graphite surface. The experiment was conducted in hydrogen and deuterium plasmas where the ion mean free path is 5–10 times larger than the ion gyro radius and with varying magnetic field angle ψ from 0° to 90° normal to the target surface. The hydrogen ions actively reacted with carbon, leading to the formation of conical tips with axes directed along the incident ion flow direction. The ion incident angle was measured from the etched graphite images taken by scanning electron microscopy. The measured angles were compared to those calculated using Ahedo's fluid magnetic sheath model. In addition, we adopted the nominal Bohm criterion at the electrostatic sheath edge due to the larger ion gyro radius than the sheath. The results show that the ion incident angle was inclined to the normal direction with respect to the magnetic field angle because of the effect of ion collisions on ion motion in the presheath. The collisional effect on the ion motion is drastic for an oblique magnetic field angle ψ > 85°. This study demonstrates that the collisional property of the ions is crucial to guide the ion motion in magnetic (pre)sheath and to determine the ion incidence angle at the surface, even in collisionless and weakly magnetized plasmas.

Understanding the interaction between magnetized ions and material surfaces is crucial in the design of magnetic fields facing target structures. The primary objective is to reduce wall material loss while simultaneously mitigating the production of impurities reintroduced into the plasma through ion sputtering. This approach also leverages the preservation of the plasma state and extends the lifespan of facing structural materials in fusion reactors, magnetron sputters, and Hall thrusters.1–3 

The ion sputtering yield is significantly affected by the incident angle of magnetized ions, raising the question of whether the incident angle of ions on the material surface aligns with the trajectories of magnetic field lines, especially when faced with an oblique magnetic field orientation relative to the material surface. This prompts a thorough investigation of ion motion within the presheath and sheath regions, where magnetic field angles vary in magnetized plasmas.4 

Chodura5 conducted an examination of sheath structures, exploring a range of magnetic angles while not considering ion collision behavior. In this study, a conceptual framework is introduced that encompasses three distinct regions: the presheath, the magnetic (pre)sheath, and the Debye sheath. The magnetic (pre)sheath is defined as the region where ions, moving at the Bohm speed, transition from following the magnetic field lines to aligning themselves perpendicular to the material surface. In this work, we refer to it as the Chodura presheath (CP). Chodura argued that for plasmas with Debye sheath conditions scaled less than the ion gyroradius, CP exists between the Debye sheath and the general presheath with scaling of the ion sound speed over the ion gyrofrequency, coupled to the incident angle of the magnetic field. Stangeby6 further refined CP to be a sheath in the case of strongly magnetized plasma as the Debye sheath scaled to the ion gyroradius. The scale of CP was experimentally observed in weakly magnetized plasma through measurements of the space potential distribution.7 

Riemann investigated the collision effects on the formation of magnetic sheaths, utilizing sheath models that integrated ionization and friction with magnetic field angles.8 This investigation identified a transition region, defined by the collision mean free path, that bridges the gap between the sheath and the presheath. Within this transition region, collisions adjust the plasma distribution and ion drift velocity up to the sheath edge. However, the mechanisms governing the formation of Chodura's magnetic (pre)sheath in this transition region remain unclear.

Ahedo et al.9,10 developed a fluid sheath model to investigate the influence of ion collisions on magnetic sheath formation. The authors employed two scaling factors when simulating ion motion: one involving the ratio of the collision mean free path to the ion gyroradius, and the other relating to the Debye length and ion gyroradius for the presheath and sheath regions, respectively. Their study revealed that collisions caused ions to deviate from their incident angles relative to the magnetic field angle ψ, which is normal to the wall. Furthermore, they observed the formation of a Chodura (pre)sheath in the magnetized presheath under weak magnetic field conditions, maintaining a quasi-neutral state. To maintain neutrality and achieve a Bohm velocity in the magnetic field direction at the edge of the Chodura sheath, a preceding Chodura sheath is required. This presheath electric field is associated with the collisional effect, leading to the deviation of the incident ion angle from that of the magnetic field due to the influence of the E × B drift.

Kim et al.7 conducted experimental investigations in weakly collisional and magnetized plasma with various magnetic field angles, revealing that the presheath consists of two distinct regions: one influenced by the magnetic angle, scaled with the ion gyroradius with electron temperature (=Bohm speed over ion gyrofrequency), and the other influenced by ion collisions (scaled with ion collision mean free path). This study was limited to magnetic field angles ψ < 80° due to experimental setup challenges, thus the predicted threshold angle of the magnetic field11 has not been explored. Siddiqui et al.12,13 measured the two-dimensional ion speeds in the presheath region using laser induced fluorescence (LIF) while varying the magnetic field angle and pressure. They observed the E × B ion drift velocity in the magnetic presheath, which was inferred to be determined by the collision effect on ion motion in the magnetic presheath region. However, due to limitations in spatial resolution and the nature of two-dimensional measurements in these studies, investigating the motion of ions in the presheath and sheath, specifically the angle of ions just before hitting the wall, was challenging. The ion angle represents the accumulated results of deviation by collision in the three-dimensional ion motion in the presheath, making measurements of the angle of incident ions at the wall crucial for tracing ion motion in the presheath and sheath.

This study conducted experiments by varying the magnetic field angle to the surface normal, ψ = 0° to nearly 90°. The collision mean free path of weakly collisional and magnetized hydrogen plasma is comparable with ion gyroradius, but it is much smaller than the Debye length. To measure the ion incidence angle, hydrogen plasma and carbon etch reaction were utilized. Hydrogen ions reacted with carbon, causing carbon to erode in the direction of the incoming hydrogen ions. As ions decreased in depth, the remaining carbon took on a conical shape. Therefore, the direction of the conical tip represented the angle of incidence of hydrogen ions. The incident angle of ions was determined by analyzing scanning electron microscopy (SEM) images of sputtered graphite targets. These measured angles were then compared to the ion incident angles calculated using Ahedo's fluid magnetic sheath model.9 It is important to note that this calculation assumed that ions entered with thermal velocity along the magnetic field line in the presheath region. Ions entered the sheath with Bohm velocity in the direction normal to the target surface.

Section II introduces a sheath model for simulating ion motion in the magnetic presheath, based on Ahedo's intermediate and weakly magnetized sheath model.9 In Sec. III, the experimental setups are detailed, including the generation of hydrogen plasma in the electron cyclotron resonance (ECR) source and the alignment of the DC biased graphite target with the magnetic field at the central region of the chamber. Spatial plasma density, potential, and temperature were measured using Langmuir probe and kept constant. The target was placed in the uniform plasma region. Collision parameters were calculated. Section IV presents the measurement results of the incident ion angles, as analyzed from SEM images and compared to the predictions from Ahedo's model. It also discusses the collision effect on ion motion in the presence of a glanced magnetic field, the influence of collision on magnetic presheath formation and the threshold angle of the magnetic field. The conclusion is provided in Sec. V.

In Fig. 1, schematic diagrams are introduced for the coordinate system used to simulate ion trajectories in the magnetic presheath and electrostatic sheath formed in front of the graphite target. It represents a configuration of the incident magnetic field line to the plasma-facing wall (graphite) and the magnetic presheath developed in front of the bias wall. The ion motion in the presheath and sheath formation has been investigated with Ahedo's presheath with modified the initial condition as the ion thermal velocity along the magnetic field lines and the Bohm velocity at the sheath with the normal direction to the wall.9,10 Additionally, ion collisions are taken into account in the presheath space. Since the ion gyroradius is much larger than the sheath size, the influence of the magnetic field is not considered in the sheath region. This space is defined as the Debye sheath as shown in Fig. 1. The following is a detailed description of the simulation.

FIG. 1.

(a) The reference frame adopts a Cartesian coordinate system. (b) The configuration of the magnetic field in the presheath (transition) region and sheath region. The magnetic field with angle ψ directs normal to the x-y plane and the E × B drift motion is along z-direction in this system. Simulation based on Ahedo model is carried out in between the presheath edge and electrostatic (Debye) sheath (SE) where the ions have Bohm velocity, which is shown in (b).

FIG. 1.

(a) The reference frame adopts a Cartesian coordinate system. (b) The configuration of the magnetic field in the presheath (transition) region and sheath region. The magnetic field with angle ψ directs normal to the x-y plane and the E × B drift motion is along z-direction in this system. Simulation based on Ahedo model is carried out in between the presheath edge and electrostatic (Debye) sheath (SE) where the ions have Bohm velocity, which is shown in (b).

Close modal

In our simulation, ion motion within the sheath was modeled using the Debye sheath model. The initial velocity of ions entering the sheath was determined using the 3D ion velocity calculated at the sheath edge point obtained from the presheath, with the vertical entry velocity assumed to be the Bohm velocity. This allowed us to perform simulations at the discontinuity point. Ion simulations in the sheath can provide information about the characteristics of ions entering the target, specifically the ion entry angle. In this simulation, it will be possible to interpret ion behavior in the presheath and sheath based on the angles formed by ions entering, taking into account magnetization and collisionality. For this purpose, the simulation considered simulating ion entry angles into the target based on the strength of the magnetic field and the ratio of collisionality.

Cartesian coordinate system is adopted as shown in Fig. 1(a), and the origin Oxyz represents the presheath edge [denoted as “PE” indicated the presheath edge as shown in Fig. 1(b)]. The wall (target) surface is placed at x = x W, which is a perfectly absorbing wall and assumed to be negatively biased with respect to the ground as the bias potential ( U W < 0). It is assumed that the ion diffused with thermal velocity along the magnetic field line at origin, Oxyz, corresponds to the edge of PE in Fig. 1(b). The uniform magnetic fields are incidents with an angle, ψ, with respect to the normal direction of the wall surface, denoted as B = B ( cos ψ , sin ψ , 0 ). The angle of incident ions at the wall is denoted as ϕ in this simulation.

The motion of ions and the spatial distribution of the electric potential in presheath are simulated with the following equations: the Poisson equation, the Boltzmann relation for electrons, the continuity equation for ions with ionization, and the momentum equation along the x-direction as
d 2 U d x 2 = e ε 0 ( n e n i ) ,
(1)
k B T e ln n e e U = ( constant ) ,
(2)
d d x ( n i V x ) = ν i n e ,
(3)
m i V x d V d x = e V × B ( e d U d x + k B T i n i d n i d x ) x ̂ ν c m i V ,
(4)
where n, m, T are the density, the mass, and the temperature, and subscripts e and i, denoted below, represent electrons and ions, respectively. U is the space potential, V = ( v x , v y , v z ) is the drift velocity of ions. The single ionized ions are assumed as Z = 1. The collision frequency of ion ν c in Eq. (4) and the ionization frequency ν i in Eq. (3) set a constant value. ε 0 is the permittivity constant of vacuum, and k B is the Boltzmann constant. In the source term of Eq. (2), it is assumed that the neutral particle–electron collision for ionization reaction is dominant in the whole space. The collision term in Eq. (4) refers to the total momentum of the ion flow to be lost. It notes that the ion-neutral collision and ionization are considered. The ion–ion collision and ion–electron collision are not considered in Eq. (4), because ion–ion collision does not change drift velocity of ion flux (momentum conservation) and the momentum change and frequency of ion–electron collision momentum is too small to compare the ion-neutral collision in this low temperature plasma. The cross-section data for hydrogen plasma are taken from Ref. 14.
Equations are rewritten with the following dimensionless parameters:
Φ = e U k B ( T e + T i ) , N i = n i n 0 , v = V C s , and t e , i = T e , i T e + T i ,
(5)
where Φ is the normalized potential energy of ions e U with the thermal energy of electron k B T e, Ni, and Ne are the density of ions and electrons, respectively, the bulk plasma density n 0, v is the ion drift velocity V with Bohm velocity ( C s = k B ( T e + T i ) m i). With these parameters, Eqs. (1) and (4) can be simplified into the following form:
d 2 Φ d x 2 = ( N i N e ) λ d 2 ,
(6)
ln N e + Φ = constant ,
(7)
d d x N i v x = N e λ p .
(8)
The direction force balance equations are
( v x 1 v x ) d v x d x = v z sin ψ λ m + t i N i d Φ d x 1 λ c v x ,
(9)
v x d v y d x = v z cos ψ λ m 1 λ c v y ,
(10)
v x d v z d x = v x sin ψ λ m v y cos ψ λ m 1 λ c v z .
(11)

Here, three characteristic lengths are introduced, which are the collision mean free path of ion and neutral, λ c = C s / ν c, the ion gyroradius, λ m = C s / e B, and the Debye length, λ d = ( ε 0 k B T e / e 2 n 0 ) 1 / 2. We solve Eqs. (6)–(11) numerically and obtain the spatial distribution of ion drift velocity, densities of electrons and ions, and electric potential. In these equations, it can be understood that both the magnetic field and collisions have an influence on ion motion. Ahedo has wisely introduced a new parameter Λ c m that can trace the effect of collisionality and magnetization ratio on the ion motion.9 This parameter is adopted in our simulation. We have verified that our simulation results reproduced those of Ahedo.10,15,16

The physical characteristics of sheath and presheath regions are investigated with the measurement of those scale lengths. For electrostatic sheath, under the SE in Fig. 1(b), the Debye length appropriates as the characteristic length scale where the highest electric field builds in the normal direction to the biased wall due to only ions in electrostatic sheath. For the presheath, however, both collision and magnetization affect ion dynamics and so the characteristics of ion motion in presheath region are investigated with λ c and λ m. When the collision mean-free-path λ c . is comparable with the gyroradius λ m, presheath is a weakly magnetized presheath where λ c is appropriated for the characteristic length scale of the region. The magnetization and collisional effect on ion dynamics in presheath are not distinctive, and the collisionality parameter Λ c m is introduced as the following relation (same as Ahedo model):
Λ c m = λ c / λ m .
(12)

For the unmagnetized or collision dominated presheath, simulation is carried with Λ c m 0, while a magnetized and collisionless presheath investigates with Λ c m . This model is more suitable for the weakly collisional and magnetized case, and the simulation is carried out by varying the collision parameter Λ c m.

To investigate the magnetized and weakly collisional effect on ion dynamics in presheath, the simulation is carried out with a variation of finite Λ c m for collisionless ( Λ c m= 100) to mild collisional condition ( Λ c m= 1). Equations (9)–(11) can be further simplified using η = x / λ c . eff,
Φ = t e ln N ,
(13)
d N d η = N v x ( λ c . eff λ p d v x d η ) ,
(14)
( v x 1 v x ) d v x d η = Λ c m v z sin ψ v x λ c . eff λ p v x ,
(15)
v x d v y d η = Λ c m v z cos ψ v y ,
(16)
v x d v z d η = Λ c m ( v x sin ψ v y cos ψ ) v z .
(17)
From Eq. (15) of the x-directional ion motion in the perpendicular direction to the bias wall as indicated in Fig. 1, this equation has a singularity at v x = 1, which corresponds to the sheath edge. Then the electrostatic sheath model is replaced to calculate the ion motion, which is described in Sec. II B. The ion velocity vy obtained from Eqs. (16) and (17) represents the ion flow resulting from E × B drift motion. It notes that the electric field Ex in presheath is dominantly generated in the x-direction along with the collisional plasma diffusion to the wall.

Using these equations, we can observe the influence of collisions and magnetic fields on ion velocity changes in the pre-sheath. In this study, the equations were solved using the fourth-order Runge–Kutta method in MATLAB. Considering that ions enter the presheath with thermal velocity in the direction of the magnetic field, the influence of initial velocity on the ion velocity at the boundary between the presheath and sheath is negligible. Therefore, the ion velocity at the point of entry into the sheath is determined by the magnitude of ion velocity changes within the presheath, corresponding to the ion velocity changes within the presheath. In Fig. 2, collision and magnetic field angle effect on ion motion in the presheath is investigated, with a simulation of ion drift velocity satisfying Vx = Cs at the SE and varying the collisional parameter Λ c m with changes in the magnetic field angle. Figure 2(a) represents the ion velocity, which is guided by the magnetic field. The vy velocity represents the ion velocity change due to collisions within the pre-sheath and magnetic behavior. When increasing Λ c m, the variation of the ion velocity at the sheath edge increases with the incident angle ψ of the magnetic field. This is because a large Λ c m implies that the ion collision mean free path is longer than the ion gyroradius, making ion collisional diffusion more effective on the y-directional motion. When the magnetic field becomes glancing to the target as ψ becomes 90°, the result shows that the collisional diffusive ions enter the sheath drastically enter into the sheath. Figure 2(b) shows the Ex×B drift ion motion with varying Λ c m and ψ. The vz reflects the magnitude of the E×B ion drift velocity, which is influenced by the vertical electric field formed by plasma diffusion in the x direction and the magnetic field in the target horizontal plane. When increasing the collision mean free path, the electric field generated by charge diffusion becomes smaller. It is observed from Fig. 2(b) that vz decreases with increasing Λ c m. When ψ becomes 90°, E × B ion drift along the z-direction is dominated and decreases vy. In addition, vx represents the diffusive ion velocity.

FIG. 2.

(a) Ion drift velocity at the SE (Vx = Cs) in the y axis (B-field sine direction) and (b) in the z direction, which represents Ex×B drift motion with varying magnetic field angle ψ. Λ c m=100 represents nearly collisionless, magnetized plasma, and Λ c m=1 is for the case of highly collisional magnetized plasma.

FIG. 2.

(a) Ion drift velocity at the SE (Vx = Cs) in the y axis (B-field sine direction) and (b) in the z direction, which represents Ex×B drift motion with varying magnetic field angle ψ. Λ c m=100 represents nearly collisionless, magnetized plasma, and Λ c m=1 is for the case of highly collisional magnetized plasma.

Close modal

For our condition, the sheath size is much smaller than the ion gyroradius, and the sheath can be assumed to be an electrostatic sheath or a Debye sheath. The Debye sheath model is adopted in this work, and the magnetic field effect is ignored.9 Simulation is carried out with the ion entrance velocity as vx is Bohm velocity, and vy and vz are obtained in Fig. 2.

Equations. (9)–(11) are rewritten using a normalized variable ξ = x / λ d and d x = λ d d ξ, providing Eqs. (17) and (18). For a low pressure and weakly magnetized dense plasma, where the Debye length λ d is much smaller than λ c, λ p, and λ m; therefore, λ d / λ c , m 0. The Gauss law and the equation of motion in the sheath follow as
d 2 ϕ d ξ 2 = ( N i N e ) ,
(18)
( v x 1 v x ) d v x d ξ = t i d ϕ d ξ , d v y d ξ = 0 , and d v z d ξ = 0.
(19)

This method of calculation cannot simulate the potential spatial distribution because the electric field is calculated as ∼0 at the initial value of the sheath edge. However, under the condition that the small Debye length assumption holds, if only the potential difference is accurately simulated, the energy and direction of the ion flux reaching the wall can be calculated. The model sets the potential difference applied from the bulk to the wall. It ends the calculation at the point where the spatial potential reaches the set value. It is capable of simulating the sheath potential of the experimental conditions.

The simulation model tracking ion motion in the presheath (described in Sec. II A) and sheath (Sec. II B) in the presence of collisions in the magnetized plasma has been completed. Ions, which have thermal motion in the direction of the magnetic field, enter the presheath, moving through the Debye sheath, and then ion incidents to the target. It allows us to predict the incident ion angles with varying collisionality and the diffusion characteristics of ions with respect to the magnetic field angle. The ion angles are presented along with experimental measurement data in Fig. 6.

To investigate the impact of magnetic fields and collisionality on ion motion in the presheath and sheath regions, the incident angles of ions at the target surface were measured while varying the magnetic field angle. The target was positioned in a region with a uniformly aligned magnetic field, and the incident magnetic field angle was adjusted by tilting the target. Graphite, known to undergo etching reactions in hydrogen plasma, was chosen as the target material, and a DC bias was applied to the sample (Sec. III A). The etching reactions caused by incoming hydrogen ions led to the formation of graphite tips. It was therefore assumed that the primary direction of the graphite tip indicated the direction of the incident ion. The size of the tip was less than a μm, and the tip angles were measured through SEM image analysis (Sec. III B). In Sec. III A, the experimental apparatus and setup for measuring ion entry angles will be introduced.

The experiments were conducted in an electron cyclotron resonance plasma source at Seoul National University (SNU-ECR), as illustrated in Fig. 3. The setup includes a plasma generation region where the electron cyclotron resonance heating zone is formed and a downstream region where the graphite target is positioned. A pair of current coils in the source chamber generates a magnetic field exceeding 1000 Gauss, and a 2.45 GHz microwave is transmitted through a waveguide. In the region with 875 Gauss in the source chamber, the resonant acceleration of gyrating electrons by the polarized electric field leads to the generation of dense plasma, which then diffuses into the downstream chamber. The graphite specimen, placed on the target holder (referred to as the graphite target), is negatively DC biased at −100 V. The sheath thickness, estimated based on the Child law model, is approximately 100 μm, significantly smaller than the ion collision mean free path and ion gyroradius.

FIG. 3.

Schematic diagram of the ECR device (left) and pictures of the graphite target holder set at the angle of 80° to the normal of the support plane, corresponding to the magnetic field angle ψ = 10° in the picture (right top). Four graphite specimens are placed in four holes on the target holder, and data are obtained by averaging the measured values.

FIG. 3.

Schematic diagram of the ECR device (left) and pictures of the graphite target holder set at the angle of 80° to the normal of the support plane, corresponding to the magnetic field angle ψ = 10° in the picture (right top). Four graphite specimens are placed in four holes on the target holder, and data are obtained by averaging the measured values.

Close modal

In an ECR plasma source, the magnetic field is not uniformly generated, and the plasma diffuses along the field line, resulting in non-uniform ECR plasma and magnetic field distribution in the chamber. To mitigate experimental uncertainties arising from magnetic field divergence and energetic electron flux, the graphite target holder was positioned at the center of the downstream chamber, 3 cm away from the source chamber. This setup minimizes hot electron damage to the target. The magnetic field strength is approximately 450 Gauss and nearly uniform, with a perpendicular magnetic field angle and field divergence of less than 1°. This configuration effectively addresses experimental ambiguities since the observation area of the graphite target is on a microscale, much smaller than the scale of magnetic field divergence and non-uniform plasma. Plasma density and temperature near the target were measured using Langmuir probes (1.5 mm length, 0.2 mm diameter, cylindrical, hand-made). The image in the top right corner of the figure depicts the configuration of the graphite target holder with a magnetic field incident angle of 10°. The image in the bottom right corner provides a frontal view of the installed target. Measurements were conducted using four graphite samples, and the results were averaged from these samples for analysis.

Experiments were carried out using hydrogen and deuterium plasmas with the operating pressure of 2.5 and 5 mTorr (the base pressure was below 2 × 10−6 Torr) and microwave power kept at 400 W. Electron density, ne, and electron temperature, Te, measured by a Langmuir probe, were 2.1 × 1017 m−3 and 5.0 eV (2.5 mTorr of H2), 3.0 × 1017 m−3 and 4.5 eV (5 mTorr of H2), and 2.3 × 1017 m−3 and 5.1 eV (2.5 mTorr of D2), respectively.

It is known that several ion species as H+, H2+, H3+, and negative ions are generated in the ECR hydrogen plasma. The concentration of those ion species was calculated by using a global balance model17 for hydrogen plasma. Based on the simulation results, the dominant ion species is H2+(D2+) is chosen for the given experimental condition, and only the collision with hydrogen molecule ions, H2+–H2, is considered in this work. LIF measurement experimental data obtained Ti ∼ 0.1 eV under similar conditions of 2.5 mTorr in ECR,18,19 adopted to obtained the momentum collision cross-section for H2+–H2, 1.91 × 10−18 m2. It notes that the value of collision frequency represents momentum transfer collisions of hydrogen molecule ion and neutral hydrogen assumed to be twice the charge exchange cross section.16 It is estimated that the ionization rate coefficient with Te = 5 eV for e + H2 → H2+ + 2e is 7.6 × 10−16 cm−3/s.16 The characteristic lengths of experimental condition are summarized in Table I. The electronegativity resulting from H- ions in low pressure condition under 5mTorr is very low,20 and it was found to be less than 0.5% in the global model,17 indicating its negligible impact, and therefore, it was not considered in this study.

TABLE I.

Characteristic lengths to adopt in the experiment, which are obtained with H2+(D2+) ions and H2(D2) neutral particle.

Characteristic length 2.5 mTorr H2 5 mTorr H2 2.5 mTorr D2
λ c  Collision MFP  v i . t h / ν c  14.1 mm  7.4 mm  14.5 mm 
λ m  Ion Gyroradius  ( m v i . t h ) / ( e B )  1.4 mm  1.4 mm  2.8 mm 
Λ c m  Collisionality  λ c . eff / λ m  10  5.3  5.2 
λ D  Debye length  ε 0 k B T e / e 2 n 0  39 μ 28 μ 35.3 μ
Characteristic length 2.5 mTorr H2 5 mTorr H2 2.5 mTorr D2
λ c  Collision MFP  v i . t h / ν c  14.1 mm  7.4 mm  14.5 mm 
λ m  Ion Gyroradius  ( m v i . t h ) / ( e B )  1.4 mm  1.4 mm  2.8 mm 
Λ c m  Collisionality  λ c . eff / λ m  10  5.3  5.2 
λ D  Debye length  ε 0 k B T e / e 2 n 0  39 μ 28 μ 35.3 μ

As indicated in Table I, the collision mean free path is approximately 10 mm, and the ion Larmor radius falls within the range of 1–2 mm. The experiments were conducted under varying collisionality conditions in the simulation, ranging from 5.2 to 10. The effects of collisions within the range of this Λ c m on the ion velocity at the sheath edge and changes in the potential and length of the presheath are specified in Fig. 2 (Sec. II) and Fig. 7 (Sec. IV). Given the extremely small size of the Debye sheath, on the order of micrometers, it becomes impractical to measure experimentally. Furthermore, since the Debye length is significantly smaller compared to the collisional mean free path and the ion gyroradius, it is reasonable to neglect collision effects in the magnetic field and consider the electrostatic sheath ion momentum equation, as described in Eq. (19).

In the experiment, the impact of ion collisionality and magnetic fields on ion behavior was analyzed by comparing the ion entry angles predicted by simulations with the values measured on the samples. The etching results of a graphite target (diameter 10 mm, thickness 3 mm, cylindrical, made in Korea) by hydrogen ions, which create directional tips, were used to measure the ion entry angles into the target.21 This method provided a comprehensive analysis of how ion behavior is affected by collisionality and magnetic fields.

Direct measurements of the ion incident angle of hydrogen ions were conducted using the graphite target, which exhibits a high sputtered etch rate with hydrogen atoms, leading to the formation of conical-shaped tips. Figure 4 shows SEM(SUPRA in SNU NICEM) images of (a) the pristine target, (b) the target irradiated with helium, and (c) the target irradiated with hydrogen ions, with the normal magnetic field direction set at 0°. Figures 4(a) and 4(b) do not show tip formation. However, exposure of the graphite to hydrogen plasma results in vigorous etching through physical and chemical reactions involving energetic hydrogen ions and neutral hydrogen, leaving behind nano-scaled conical tips on the graphite surface, as depicted in Fig. 4(c). It is important to note that the SEM image was captured with a tilt angle with respect to the surface normal, allowing for clear visualization of the conical tips. It is assumed that the summit of the tip is amorphous and remains so during etching, preserving the conical shape of the graphite. The mechanism behind the formation of these tips is not fully understood,22–25 and the detailed analysis of tip formation is beyond the scope of this study. It is hypothesized that the direction of the tip axis is parallel to the incident angle of the hydrogen ions. With an increase in ion flux in the normal direction, the height of the tip decreases with fewer tips formed, but the angle remains unchanged.

FIG. 4.

Scanning electron microscope (SEM) images of the surface of (a) pristine, (b) helium ion irradiated, and (c) hydrogen ion irradiated graphite targets. Images are taken with the tilt angle of 30° with respect to surface normal direction.

FIG. 4.

Scanning electron microscope (SEM) images of the surface of (a) pristine, (b) helium ion irradiated, and (c) hydrogen ion irradiated graphite targets. Images are taken with the tilt angle of 30° with respect to surface normal direction.

Close modal

In this study, the angle of the conical tips is measured from SEM images. Since the SEM images are obtained with a tilt angle of 30°, the analysis of the incident angle of ions takes into account the tilted angle of the image. This approach necessitates orthogonal images captured from two different angles. Therefore, SEM images were taken from two directions to facilitate this analysis.

In Fig. 5, the analysis of angle of incident ions from two SEM images is introduced. Figure 1 shows an image with a 30° tilt with respect to the normal direction of the target surface. Because the ion incidental direction is three-dimensional with respect to the magnetic field line, which represents the surface normal direction and the E × B direction, the SEM images of nano-tips were taken in two orthogonal directions as shown in Fig. 5. First, in Fig. 5(a), the cubic model represents the domain of the graphite tip, which is formed in the ion incident direction. Figure 5(a) describes the method of measuring the angle of the tip θ tip formed in the (a, b, c) direction. The two seeming tilt angles of nano-tips, θ 1 and θ 2, were measured to calculate the morphological tilt angle of the nano-tips, θ tip. θ 1 and θ 2 are the angles measured in image (1) transferred to the plane in which the graphite plane (x-y plane) is rotated 30° on the y-axis, and in image (2) transferred on the plane rotated by 30° on the x-axis. Assuming that the direction vector of the nano-tip is ( a , b , c ), vector elements a and b are functions of θ 1, θ 2, and θ SEM = 30 ° as
{ tan θ 1 ( 3 a / 2 + 1 / 2 ) = b , tan θ 2 ( 3 b / 2 + 1 / 2 ) = a .
(20)
FIG. 5.

Scanning electron microscope images taken in two orthogonal directions. (a) Image showing which direction the camera is shooting from in a fixed tip, (b) and how it was actually shot on the SEM device. (c) SEM image taken with 30° tilted angle. (1) is the transcribed image of the red plane and (2) is the transcribed image of the blue plane in figure (a). The tilt angle of nano-scaled tips, θ tip, can be obtained from the seeming tilt angles, θ 1 and θ 2, that are taken at two orthogonal imaging directions.

FIG. 5.

Scanning electron microscope images taken in two orthogonal directions. (a) Image showing which direction the camera is shooting from in a fixed tip, (b) and how it was actually shot on the SEM device. (c) SEM image taken with 30° tilted angle. (1) is the transcribed image of the red plane and (2) is the transcribed image of the blue plane in figure (a). The tilt angle of nano-scaled tips, θ tip, can be obtained from the seeming tilt angles, θ 1 and θ 2, that are taken at two orthogonal imaging directions.

Close modal
Here, the third vector element c is set to be unity. By solving Eq. (21), the two vector elements, a and b, can be obtained, and the three-dimensional tilt angle of nano-tips, θ tip, can be obtained as
a ( θ 1 , θ 2 ) = tan θ 2 ( 3 tan θ 1 / 2 + 1 ) 2 1.5 tan θ 1 tan θ 2 ,
(21)
b ( θ 1 , θ 2 ) = tan θ 1 ( 3 tan θ 2 / 2 + 1 ) 2 1.5 tan θ 1 tan θ 2
(22)
θ tip ( θ 1 , θ 2 ) = tan 1 ( a 2 + b 2 ) .
(23)
Here the qTip represents the ion incident angle at the graphite introduced as f of Figure 1.

We compared the measured ion incidence angles from conical tips formed on graphite to the simulation results in Fig. 6. Figure 6(a) presents the measured ion incident angles obtained at magnetic field angles of 20°, 40°, 60°, 80°, 85°, and nearly 90° under the condition of a −100 V target bias, forming a thick electrostatic sheath. Figure 6(b) represents the ion incident angles measured at magnetic field angles of 20°, 40°, 60°, and 80° with the target grounded, forming a thin electrostatic sheath. The measured ion incident angles were compared with the simulation values obtained from Fig. 2, as shown by solid lines. These results illustrate the influence of the sheath size on the ion incident angle.

FIG. 6.

(a) Ion incident angles measured at various magnetic field angles under biased wall condition (−100 V) and corresponding calculated values from the model. Closed black squares indicates the experimental results taken for 2.5 mTorr (hydrogen plasma), the red squares are for 5 mTorr (hydrogen plasma), and the blue squares are for 2.5 mTorr (deuterium plasma). The dotted and solid lines represent the simulation results with various Λcm. (b) Ion incident angles measured at different magnetic field angles under grounded wall condition and corresponding calculated values from the model. Closed black squares indicate the experimental results taken for 2.5 mTorr (hydrogen plasma), the red squares are for 5 mTorr (deuterium plasma). (a) and (b) demonstrate the influence of the sheath size on the incidence angle of ions.

FIG. 6.

(a) Ion incident angles measured at various magnetic field angles under biased wall condition (−100 V) and corresponding calculated values from the model. Closed black squares indicates the experimental results taken for 2.5 mTorr (hydrogen plasma), the red squares are for 5 mTorr (hydrogen plasma), and the blue squares are for 2.5 mTorr (deuterium plasma). The dotted and solid lines represent the simulation results with various Λcm. (b) Ion incident angles measured at different magnetic field angles under grounded wall condition and corresponding calculated values from the model. Closed black squares indicate the experimental results taken for 2.5 mTorr (hydrogen plasma), the red squares are for 5 mTorr (deuterium plasma). (a) and (b) demonstrate the influence of the sheath size on the incidence angle of ions.

Close modal

Figure 6(a) illustrates the variation of the collisional mean free path and Larmor radius as the ion incident angle changes. The measured ion incident angles (dots) as a function of magnetic field angle and the calculated ion incident angles (lines) are separated into presheath and sheath regions for different magnetic field angles. The error bars on the measured values represent the standard deviation of the angles measured from a random sampling of 60 tips from graphite SEM images. Figure 6(a) presents the ion incident angles measured on a wall with a −100 V DC bias compared to the plasma potential measured by a Langmuir probe. The black, red, and blue dots represent ion incident angles measured with a graphite material probe under hydrogen conditions of 2.5, 5, and deuterium conditions of 2.5 mTorr, respectively.

In Fig. 6(a), the black dashed line represents the calculated result assuming no collisional effects on ion behavior, while the black solid line and red solid line represent ion incident angles calculated with Λcm = 10 and Λcm = 5, respectively. Figure 6(b) shows the results measured on a grounded wall, where the black and red dots represent ion incident angles measured under hydrogen conditions of 2.5 and 5 mTorr, respectively, from the eroded graphite material. The black solid line and red solid line represent ion incident angles calculated with Λcm = 10 and Λcm = 5, respectively. The solid lines indicate the calculation for both presheath and sheath regions, while the dashed line represents the calculation that stops at the presheath with all potential differences accounted for in the presheath. The blue dotted line represents the reference where the magnetic field angle equals the ion incident angle.

The results in Fig. 6(a) for the −100 V DC bias condition demonstrate good agreement with the predictions of the Λcm = 5 and 10 simulations for magnetic field incident angles up to 80°. The variation in ion incident angle observed in Fig. 6(a) with pressure and ion species illustrates the definition of collisionality relative to magnetization in the presheath. The results for the 2.5 mTorr hydrogen plasma condition, where the ion mean free path (MFP) is 14.1 mm and the Larmor radius is 1.4 mm, denoted by black dots, follow the Λcm = 10 line. In comparison, the 5 mTorr hydrogen plasma condition, where the neutral particle density increases, corresponds to a condition where the ion MFP is reduced by approximately half as the density of neutral particles increases. The 2.5 mTorr deuterium plasma condition exhibits collisional characteristics nearly identical to the 2.5 mTorr hydrogen plasma. However, given that the ion mass is doubled, the Larmor radius also doubles. Although the variables for the two conditions, MFP and Larmor radius, differ, the measured ion incident angles for both conditions closely follow the simulation results with Λcm = 5. This outcome shows that the characteristics of collisions are represented by the ratio of gyromotion to collision in the magnetic plasma boundary.

The −100 V target bias allowed ions to have Bohm velocity at the presheath and losses to the sheath, thanks to sufficient diffusion and loss to the sheath. If ion losses to the sheath are not significant, the potential in the presheath increases by collision, making it difficult to establish conditions with Bohm velocity. This phenomenon is observed in the experimental condition with the target grounded in Fig. 6(b). For Λcm = 5, the sheath boundary is not observed for magnetic field angles above 40 (see the dashed line and further explanation that follows), and for Λcm = 10, as MPF increases relatively, the sheath boundary is not observed around the magnetic field angle of 60. These results demonstrate that the increase in presheath potential by the magnetic field angle on ion behavior in the presheath is manifested by ion losses due to collisions in the presheath. Therefore, in weakly collisional, magnetized plasma, ion behavior in the presheath is affected by the magnetic field's gyromotion and collisions and the 3D behavior of ions lost from the presheath to the sheath, depending on the magnetic field angle. Consequently, when the magnetic field incident angle is different from the ion incident angle, and the ion collisional MFP is comparable with the ion's Larmor radius, the ion incident angle converges to the magnetic field incident angle.

For the biased wall condition and the ground wall condition, the Debye sheath (DS) size calculated using the Child–Langmuir sheath model is (a) 245 μm (2.5 mTorr), 176 μm (5 mTorr) and (b) 59 μm (2.5 mTorr), 38 μm (5 mTorr). These sizes are smaller than both the mean free path (MFP) and gyroradius, so the ions accelerate to wall normal direction in the electrostatic sheath. The simulation assumed Vx = Cs condition at the sheath boundary, as introduced in Sec. II. When changing the magnetic field incident angle from 0° to 90° under the biased wall condition, it is shown that the ion incident angle remains below 25°. This demonstrates that the ions directly accelerate toward the wall perpendicular direction in the sheath, thus incident angle is significantly different from the magnetic field incident angle in the sheath with a strong electric field. On the other hand, the potential difference between the plasma and the wall is measured to be 12–13 V (2.4–2.6 Te) in the ground target condition. Under the grounded wall condition, the ion incident angle measured is higher compared to the biased wall condition when the magnetic field angle is increased because of weak sheath potential. This indicates that in the electrostatic sheath (λD ≪ λm, λc), ions are accelerated toward the wall.

The change in ion incident angle due to collisionality shows contrasting trends between biased wall and grounded wall conditions. In the case of a biased wall, when collisionality increases, the ion incident angle is measured to be lower. Conversely, in grounded wall conditions, when collisionality increases, the ion incident angle is measured to be higher.

The reduction in ion incident angle with increased collisionality in biased wall conditions is attributed to the collisional effect on gyromotion in the presheath region, leading to a change in y- and z-axis velocities at the sheath edge (Fig. 2). As simulated in Fig. 2(a), Vy at the sheath boundary increases following the magnetic field angle. However, it is evident that as the magnetic field angle exceeds 80° (becoming nearly parallel to the target), Vy decreases sharply. This indicates that under glancing angle conditions, the motion of ions is primarily characterized by the inability of ions to undergo a transition in the y-axis direction due to the magnetic field. Instead, diffusion and acceleration along the B-field perpendicular direction dominate, following the magnetic field flux. Figure 2(b) shows that the ExB ion drift velocity decreases as collisionality decreases but sharply increases as the B-field incident angle becomes larger. The electric field imposed on the presheath weakens in the presence of lower collisions in an oblique magnetic field angle, but the presheath E-field in a low collisional condition sharply increases in a glancing angle condition (Fig. 7). These results indicate that ion behavior in the presheath is influenced by collisions and the magnetic field angle under weakly collisional and magnetized conditions, which is supported by experimental results.

FIG. 7.

(a) Electric potential drop and (b) length of presheath (initial state to Vx = Cs) according to the collision and magnetic field angle change calculated by the weakly collisional and magnetized presheath model.

FIG. 7.

(a) Electric potential drop and (b) length of presheath (initial state to Vx = Cs) according to the collision and magnetic field angle change calculated by the weakly collisional and magnetized presheath model.

Close modal

In both biased and grounded wall conditions, the ion behavior in the presheath is identical, with the key difference between the two conditions occurring in the electrostatic sheath. The sheath potential difference is influenced by collisionality, which alters the x-axis direction velocity at the wall.

In scenarios with higher collisionality, more ions transfer their kinetic energy to neutral particles in the presheath. Consequently, they require more potential energy to reach the Bohm condition, leading to an increase in presheath potential in more collisional conditions [Fig. 7(a)]. When the potential difference between the plasma and the wall is constant, the increase in presheath potential results in a decrease in the sheath potential. In biased wall conditions, this change results in a minor variation, within 3% of the total sheath potential (17–19 Te). In contrast, in grounded wall conditions, this change leads to a substantial shift, accounting for at least 15% or more (0–1.5 Te). As a result, in grounded wall conditions, the reduction in x-axis velocity due to the decrease in sheath potential caused by increased collisionality has a more significant impact than the reduction in y- and z-axes velocities, leading to an increase in ion incident angle as collisionality rises.

In summary, collisions disrupt gyromotion in the presheath, reducing the y- and z-axes velocities of ions reaching the sheath edge. Additionally, collisions increase the presheath potential drop under oblique angle conditions (less than 80°), further influencing ion behavior.

The measured ion incident angles closely align with the calculated results under oblique angle conditions (less than 80°). However, for biased wall conditions with magnetic angles greater than or equal to 85° (glancing angles), the measured ion incident angles significantly deviate from the collisionality line in all conditions. The increase in ion incident angle for biased wall conditions at higher magnetic angles might be due to the effective reduction in collision effects, resulting in higher y- and z-axes velocities at the sheath edge or lower sheath potential, leading to a decrease in x-axis velocity.

The experimental approach involves manipulating the target's angle within consistent plasma conditions while maintaining collisionality proportional to the neutral gas density. However, at glancing angles (85° or more), the observed variation is expected to stem from distinct potential distributions at the plasma boundary compared to the two-layer model used in the study, indicating a limitation in the model's interpretive capacity. In scenarios where a more robust potential is established in the presheath than predicted under glancing angle conditions, the y-axis velocity is not expected to undergo pronounced changes in a glancing angle condition. Conversely, the E×B ion drift in the z-axis direction is influenced by the strength of the presheath electric field. If the electric field in the presheath of a glancing angle condition is much lower than the model's results, the z-axis velocity at the sheath edge should decrease. Simultaneously, the increase in presheath potential induces a proportionate decrease in sheath potential, resulting in a decrease in x-axis velocity near the wall. Consequently, as supported by the experimental findings, the ion incident angle experiences an increase.

This study focused on directly measuring ion incident angles in weakly collisional and magnetized plasmas, comparing these measurements with values from the Ahedo's fluid model to explore the contact angle of incident ions on biased and grounded targets. The results indicated good agreement between the experimental data and simulations for various magnetic field incident angles up to 80° under a −100 V DC bias condition. The variation in ion incident angles with pressure and ion species highlighted the role of collisionality relative to magnetization in the presheath. The comparison between different plasma conditions demonstrated the impact of collisional effects on ion behavior and the importance of considering gyro motion and collisions in magnetized plasma boundaries.

The analysis of ion incident angles under biased and grounded wall conditions revealed contrasting trends in ion behavior with increasing collisionality. In biased wall conditions, higher collisionality led to lower ion incident angles, attributed to collisional effects on gyro motion in the presheath. On the other hand, grounded wall conditions showed an increase in ion incident angles with higher collisionality due to changes in sheath potential and ion acceleration toward the wall. Under typical conditions where the magnetic field angle is not perfectly parallel to the wall, the behavior of ions in front of the wall was found to be sensitive to the magnitude of the sheath electric field.

The results also highlighted the influence of collisionality and magnetic field angle on ion behavior in the presheath, emphasizing the interplay between collisions, magnetic fields, and sheath potential in weakly collisional and magnetized plasma. The experimental findings supported the theoretical predictions, showcasing the complex dynamics of ions in the presheath and sheath regions under varying plasma conditions. For fusion plasmas, the ion dynamics are significantly influenced by the ion-ion for a dense fusion plasma and the ion-neutral collision for a gas puffing condition. Additionally, local ionization should be considered.

This work was supported by the National Research Foundation of Korea (NRF), funded by the Ministry of Science & ICT (Grant No. RS-2022-00156289), and the ITER Technology R&D Programme. In memory of Professor Noah Hershikowitz. This research is supported by the Brain Korea 21 FOUR Program (No.4199990314119).

The authors have no conflicts to disclose.

Myeong-Geon Lee: Conceptualization (supporting); Data curation (lead); Formal analysis (lead); Methodology (equal); Writing – original draft (equal). Nam-Kyun Kim: Conceptualization (lead); Data curation (lead); Formal analysis (equal); Methodology (lead); Writing – original draft (equal). Jaemin Song: Data curation (supporting). Ki-Baek Roh: Conceptualization (supporting); Data curation (supporting). Sung-Ryul Huh: Conceptualization (supporting); Methodology (supporting). Gon-Ho Kim: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

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

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