The use of non-sinusoidal waveforms in low pressure capacitively coupled plasmas intended for microelectronics fabrication has the goal of customizing ion and electron energy and angular distributions to the wafer. One such non-sinusoidal waveform uses the sum of consecutive harmonics of a fundamental sinusoidal frequency, f0, having a variable phase offset between the fundamental and even harmonics. In this paper, we discuss results from a computational investigation of the relation between ion energy and DC self-bias when varying the fundamental frequency f0 for capacitively coupled plasmas sustained in Ar/CF4/O2 and how those trends translate to a high aspect ratio etching of trenches in SiO2. The fundamental frequency, f0, was varied from 1 to 10 MHz and the relative phase from 0° to 180°. Two distinct regimes were identified. Average ion energy onto the wafer is strongly correlated with the DC self-bias at high f0, with there being a maximum at φ = 0° and minimum at φ = 180°. In the low frequency regime, this correlation is weak. Average ion energy onto the wafer is instead dominated by dynamic transients in the applied voltage waveforms, with a maximum at φ = 180° and minimum at φ = 0°. The trends in ion energy translate to etch properties. In both, the high and low frequency regimes, higher ion energies translate to higher etch rates and generally preferable final features, though behaving differently with phase angle.

Plasma etching of dielectrics such as SiO2 is an integral process in micro- and nanometer scale electronics fabrication.1 The etching of high aspect ratio (HAR) features is becoming an increasingly critical process due to the development of three-dimensional structures such as 3D-NAND memory that requires vias (circular holes) through as many as 256 alternating layers of SiO2 and Si3N4 with a total aspect ratio of up to 100.2,3 (Aspect ratio, AR, is the height of the feature divided by its width. AR > 10–20 would be considered HAR.) Plasma etching of HAR features requires highly energetic ions onto the wafer, which arrive with a nearly normal angle. This anisotropy is achieved by acceleration of the ions through the electric field in the sheath bounding the wafer and normal to the surface. The majority of HAR plasma etching of dielectrics is performed in capacitively coupled plasmas (CCPs) sustained in fluorocarbon gas mixtures, typically using multiple radio frequency (RF) power supplies.4,5

Recent industry trends have favored the use of very low frequency (VLF) biases. VLFs, typically frequencies below 1–2 MHz, correspond to the thin sheath limit in which the ion entering the sheath will cross the sheath in a small fraction of the RF cycle. In this regime, maximum ion energies incident onto the wafer extend to VRF–VDC, where VRF is the amplitude of a sinusoidal bias and VDC is the negative self-bias on the blocking capacitor. At the same time, the use of voltage waveform tailoring (VWT) is being investigated to customize those ion energy distributions. VWT consists of a bias which is the sum of multiple harmonic frequencies on a single electrode. Due to the added technical complexity and lack of fundamental understanding of its role in modifying ion and electron dynamics in and near the plasma sheath, VWT is only now being broadly adopted by industry.6–12 

The DC self-bias, VDC, is a voltage appearing on a blocking capacitor in series with the RF power supply. VDC for conventional waveforms results from the requirement to balance positive and negative currents to both electrodes on a cycle averaged basis. These currents may not naturally balance due to differences in ion and electron mobility. This asymmetry may also be the result of geometry (differences in the areas of powered and unpowered surfaces), material (for example different electron emission coefficients), magnetic field configuration (different electron mobilities adjacent to powered and unpowered surfaces) and the use of asymmetric, non-sinusoidal, voltage waveforms for power coupling producing the electrical asymmetry effect (EAE). Through control of VDC, control of the sheath potential and subsequently ion energies can be achieved.10,13,14

The direct measurement of ion energies during industrial plasma etching processes is usually not performed due to the added technical complexity or to avoid disturbing the plasma. As a result, VDC is frequently used as a proxy for ion energy onto the wafer. This is a good approximation if the sheath is collisionless and the system is operating in the thick sheath limit, where the ion requires many RF cycles to cross the sheath. For these conditions, the ion energy distribution (IED) striking the wafer has a narrow width in energy centered on −VDC. This is typically the condition intended for control of the IED through VWT.

In this paper, we discuss results from a computational investigation into the relationship between VDC and the IED for capacitively coupled plasmas employing VWT over a large range of fundamental bias frequencies. The computed reactive fluxes to the substrate and their energy distributions were used as input to a profile simulator to assess the influences of these reactive fluxes on HAR features etched into SiO2. We found that at very low frequencies, the ions are able to dynamically react to the applied potential, and as a consequence, incident ion energy onto the wafer can decouple from the trends suggested by VDC.

Brief descriptions of reactor and feature scale models employed in this investigation are presented in Sec. II. The results of the reactor scale gas phase simulations are presented and discussed for constant bias voltage amplitude in Sec. III and for constant bias power in Sec. IV. The results for simulated profiles of HAR trenches etched in SiO2 using the reactive fluxes produced by these waveforms are discussed in Sec. V. Concluding remarks are presented in Sec. VI.

The investigation of the reactor scale plasma dynamics was performed using the Hybrid Plasma Equipment Model (HPEM),15–17 which is described in detail in Ref. 18. A brief overview of the HPEM is given here. The HPEM is a two-dimensional plasma hydrodynamics model, which resolves transport phenomena using a time-slicing approach. Different physics regimes are addressed in modules that are coupled by exchanging physical quantities—electric and magnetic fields, densities, and rate coefficients. The major modules used in this work are the Fluid Kinetics-Poisson Module (FKPM), the Electron Energy Transport Module (EETM), and the Plasma Chemistry Monte Carlo Module (PCMCM). In the FKPM, the continuity, momentum, and energy equations of heavy particles are solved coincidently with Poisson's equation to provide heavy particle densities, fluxes, temperatures, and electrostatic potential. Continuity and momenta are solved for electron transport. Although the fluid treatment for ions that is used to obtain convergence is not kinetic, including continuity, momentum, and energy equations for each species does represent a non-local treatment. This non-local behavior is particularly important in determining ion transport flowing into the sheath.

The temperatures of individual neutral species and ions are computed by considering local contributions due to Joule heating (for ions), charge exchange heating for neutrals and cooling for ions, elastic collisions of electrons, exothermic and endothermic neutral reactions, and Franck–Condon heating in dissociative electron impact processes including dissociative recombination and elastic collisions between heavy particles. Non-local contributions to gas heating include convective transport and thermal conduction. The gas and ion temperatures impact the calculations through temperature dependent rate coefficients, rarefaction of gas densities, and the random thermal velocities of ions that enter the sheath, which partially determine the angular distribution of ions striking the wafer.

In the EETM, the spatially dependent electron energy distributions (EEDs) are obtained using a kinetic, Monte Carlo based approach utilizing the space and phase resolved electric fields produced in the FKPM. The HPEM accounts for two classes of electrons. Bulk electrons, whose origins are electron impact ionization, have densities that are modeled as a fluid in the FKPM with energy distributions determined by the kinetic Monte Carlo simulation in the EETM. The second class includes electrons that originate from secondary electron emission from the surfaces and subsequent acceleration in the sheath. These are typically energetic beam-like electrons whose densities and velocities are tracked kinetically. The beam electrons deposit their energy back into the system through collisions and are a source of bulk electrons once their energy falls below the minimum inelastic threshold energy for electron impact excitation.

In the EETM, the energy and angular distribution (EAD) of electrons striking the substrate are recorded. Using the EEDs in the bulk plasma, electron transport and rate coefficients are obtained, which are transferred to the other modules. After the plasma reaches a quasi-steady state in the PCMCM, the trajectories of ions and reactive neutral species are tracked using Monte Carlo techniques. Energy and angular distributions (EADs) of these species are recorded impinging onto the wafer. The reaction mechanism used for Ar/O2/CF4 plasmas is a described in Huang et al.19 The mechanism contains the following heavy particle species:

  • Rare gas: Ar, Ar(4s[3/2]2), Ar(4s[3/2]1), Ar(4s′[1/2]0), Ar(4s′[1/2]1), Ar(4P), Ar(4D), and Ar+

  • Fluorocarbon: CF4, CF3, CF2, C, CF 3 +, CF 2 +, CF+, C+, and CF 3

  • Oxygen: O2, O 2 *, O, O*, O 2 +, O+, and O,

  • Fluorine: F2, F, F*, F2+, F+, and F

  • Reaction and etch products: SiF4, SiF3, SiF2, CO, COF, COF2, CO2, FO, and CO+

Ion species impinging on surfaces generate secondary electrons based on fixed energy independent coefficients. The secondary electron emission coefficient was 0.15 for the wafer and metal and 0.05 for the dielectric covered walls.19 Excited states of argon and fluorine had a secondary emission coefficient of 0.01 on all surfaces.20 All ions recombine on surfaces to form their neutral counterparts.

A schematic of the cylindrically symmetric CCP reactor used in this investigation is shown in Fig. 1. The reactor, modeled after multi-frequency CCPs used in industrial plasma etching applications, consists of two parallel plate electrodes with a diameter of 30 cm separated by a 2.8 cm gap. A silicon wafer is mounted on the bottom electrode and a showerhead gas inlet is distributed across the top electrode. A focus ring made of quartz to improve uniformity of fluxes to the wafer surrounds the substrate. The dielectric constant of the quartz ring is ε/ε0 = 4. The conductivity of the quartz ring is negligible while that of the Si wafer is 0.05/Ω cm. The feedstock gas is an Ar/CF4/O2 =75/15/10 mixture flowing at 500  sccm through the showerhead. The reactor pressure is held constant at 40 mTorr at the location of a pressure sensor near the pump port. This is accomplished by throttling the rate of pumping.

FIG. 1.

Reactor geometry and circuit diagram.

FIG. 1.

Reactor geometry and circuit diagram.

Close modal

Voltage waveform tailoring (VWT) is provided by a power supply connected to the bottom electrode through a blocking capacitor of 100 nF. As such, we are not addressing distortion of the waveform originating from the power supply that may occur by the impedance of the transmission line and substrate. In quasi-steady state operation, the value of VDC should be independent of the value of the blocking capacitor provided that the RC time constant of the plasma-capacitor series impedance is large compared to transients in current. The capacitance used here is a balance of there being an acceptably short charging time and long enough RC time constant so that there is little variation in the VDC during the RF cycle.

The plasma is largely sustained by a sinusoidal voltage with amplitude VT applied to the top electrode with a fundamental frequency of 80 MHz. To control the dynamics of charged particles impinging on the wafer surface, a customized voltage waveform, V(t), is applied to the bottom electrode. The waveform consists of a fundamental sine wave with a frequency f0 = 1–10 MHz and four consecutive higher harmonics
V ( t ) = V 0 k = 1 N N k 1 N ( N + k ) cos ( 2 k π f 0 t + k π φ k ) .
(1)
Here, V0 is the total applied voltage amplitude, k is the harmonic order, N is the total number of frequencies, and the relative phase shift of each harmonic frequency k is φk. In this work, the number of harmonics is N = 5. The consequences of VWT were investigated by varying the phase shift of the even harmonics φ for k = 2 and 4. The resulting waveforms are shown in Fig. 2 for phase shifts of φ = 0°, 45°, 90°, 135°, and 180°. With the shape and frequency of the applied voltage being closely linked to the plasma properties, two scenarios were investigated:
  • Voltages are held constant at VT = 300 and V0 = 1500 V, as phase shift φ and fundamental frequency f0 are varied.

  • Applied voltages are dynamically adjusted to maintain a constant power deposition of PT = PB = 2000 W.

FIG. 2.

Voltage waveforms corresponding to phase shifts of φ = 0° to 180°.

FIG. 2.

Voltage waveforms corresponding to phase shifts of φ = 0° to 180°.

Close modal

The fluxes and particle energy and angular distributions to the wafer obtained from HPEM are used as input to the Monte Carlo Feature Profile Model (MCFPM) to simulate the evolution of etch features in semiconductor device materials.21,22 The MCFPM is a voxel-based, kinetic model in which pseudo-particles representing gas phase species are launched with energies and angles sampled from the distributions obtained from the HPEM. The trajectories of the incoming particles are advanced according to their initial velocity and their acceleration by electric fields produced by feature charging until a collision with a surface occurs. Due to the low density of reactive particles in a feature (typically only one particle at any given time) and long mean free path for collisions with background gases, collisions of the pseudoparticles with other gas phase species are not considered. Upon striking a surface, and based on the particle's incident energy and angle, a Monte Carlo technique is employed to determine the specific reaction that occurs. The state of the surface voxel is changed according to the reaction that is selected. The voxel can be removed (physical or chemical sputtering), replaced (chemical modification), or a voxel is added on top of the site (deposition). If the reaction yields a non-saturated or reactive gas phase species, a new gas phase pseudo-particle is launched from that site.

The mechanism for etching of SiO2 in an Ar/O2/CF4 gas mixture used in this work is described in Ref. 21. All incoming ions neutralize and deposit their charge during the first wall collision while retaining a fraction of their energy upon reflection. These particles are then called “hot neutrals.” All ions and hot neutrals striking surfaces can physically sputter all materials according to the incident particle's mass, angle, and energy as well as the surface binding energy of the material. The formation of a complex between fluorocarbon radicals and SiO2 lowers the binding surface energy, thereby reducing the chemical sputtering threshold. CxFy radicals deposit a thin polymer layer (at most a few nm), which is the primary source of sidewall passivation within the feature, whose thickness is controlled by etching with oxygen radicals.

To demonstrate the consequences of changes of ion energy and angular distributions (IEADs) resulting from varying f0 and phase angle, simulations were performed of etching a trench through 3000 nm of SiO2 covered by a 700 nm thick mask with a 100 nm opening; and terminated by a 100 nm thick Si stop layer. The total aspect ratio is approximately 30. The geometry used for the feature evolution is shown in Fig. 3. The simulation was performed in three dimensions with a mesh consisting of 60 cells wide by 750 cells tall by 20 cells deep using periodic boundary conditions, producing cubic voxels with a 5 nm side length. The etching was performed for a fixed time of 10 min.

FIG. 3.

Feature scale model initial conditions for simulating plasma etching of SiO2. PR represents photoresist.

FIG. 3.

Feature scale model initial conditions for simulating plasma etching of SiO2. PR represents photoresist.

Close modal

The intent of the simulation is a uniform trench perpendicular to the plane shown in Fig. 3 which could, in principle be performed in two dimensions. However, we have found that more robust results are produced when performing a finite depth three-dimensional simulation. In two dimensions, roughness or statistical variation in surface properties are extended infinitely into the third dimension, which can systematically prejudice the outcome, for example, overemphasizing twisting. The finite depth 3D simulations, even with periodic boundary conditions, largely removes these systemic biases.

With constant applied voltage amplitudes of VT = 300 and V0 = 1500 V, the fundamental frequency f0 was varied from 1 to 10 MHz. For each frequency, a sweep across the phase angle φ was performed with φ = 0°, 45°, 90°, 135°, and 180°. The electron density for the base case with f0 = 1 MHz and φ = 0° is shown in Fig. 4(a). The cycle averaged electron density at mid-gap is 4.5 × 1011 cm−3, sustained by an electron temperature of 3.1 eV. Ionization at mid-gap is approximately 90% by bulk electrons and 10% by secondary, sheath accelerated electrons. Electron density ne and temperature Te are fairly homogeneous as a function of radius as shown in Figs. 4(a) and 4(d). The spatial electron source distribution, generated by bulk electrons Sb, closely resembles those of ne and Te, as shown in Fig. 4(b). The smaller, but still significant amount of ionization by secondary electrons Ss is shown in Fig. 4(c). Since the bulk ionization source is mostly a consequence of electrons accelerated by sheath motion, its maximum is located at the sheath edges. Ss results from electrons accelerated by the total sheath potential and is maximum close to the bottom electrode. The dominant positive ions (Ar+, CF 3 +, CF 2 +, and O 2 +) are also shown in Fig. 4. The spatial distribution of these ions are similar, with the lower ionization potential species ( CF 3 +, CF 2 +, and O 2 +) having a broader spatial distribution to larger radii. These distributions result from charge exchange depleting Ar+ as the ions diffuses radially outward from where electron impact ionization dominantly occurs over the wafer. The neutral gas temperature (averaged over species) has a maximum value of 580 K and reactor average value of 450 K.

FIG. 4.

Plasma properties of the Ar/CF4/O2 CCP base case with VT = 300 V, V0 = 1500 V, f0 = 1 MHz, and φ = 0°. Time averaged (a) electron density ne, (b) ionization source by bulk electrons Sb, (c) ionization source by secondary electrons Ss, (d) electron temperature, Te, and ion densities, (e) Ar+, (f) CF 3 +, (g) CF 2 +, and (h) O2+.

FIG. 4.

Plasma properties of the Ar/CF4/O2 CCP base case with VT = 300 V, V0 = 1500 V, f0 = 1 MHz, and φ = 0°. Time averaged (a) electron density ne, (b) ionization source by bulk electrons Sb, (c) ionization source by secondary electrons Ss, (d) electron temperature, Te, and ion densities, (e) Ar+, (f) CF 3 +, (g) CF 2 +, and (h) O2+.

Close modal

The different voltage waveforms that result from changing the phase angle φ can alter the discharge dynamics through differences in sheath expansion heating, electron confinement and modifying surface processes such as ion and electron induced secondary electron emission. While the constant sinusoidal voltage applied to the top electrode is meant to provide a constant background ionization and electron heating, some variation in electron density ne as a function of φ nevertheless occurs and is shown in Fig. 5 for fundamental frequencies f0 = 1–10 MHz. Generally, ne is a function of the total input power at frequencies that produce electron heating. With the top electrode powered at 80 MHz, the majority of this power is expended in electron heating. With varying harmonic content with changing φ on the bottom electrode, power deposition into electrons also varies, directly translating to changes in electron heating and ionization.

FIG. 5.

Average electron density as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz at constant applied voltage.

FIG. 5.

Average electron density as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz at constant applied voltage.

Close modal

The decrease in electron density with increasing phase angle is at least partially the consequence of less efficient electron confinement. Electrons are largely confined to the bulk plasma by electric fields in the sheaths which point toward surfaces during the majority of the RF cycle. Electrons typically reach the surface (to balance positive ion current) by diffusion across the sheath during a short period during the anodic portion of the cycle. If transport by diffusion is insufficient to balance currents, an electric field reversal may occur to accelerate electrons toward the surface.23 For φ = 0° (peak waveform), the plasma sheath is nearly at its maximum width with the most negative sheath potential for the majority of the RF cycle, allowing electron transport to the surface only during the brief collapse in the sheath at the peak of the anodic portion of the cycle. The opposite is true for φ = 180° where the plasma sheath is collapsed (anode-like) for most of the RF period, allowing electrons to transport to the bottom surface for a significantly longer fraction of the cycle.

With constant voltage (VT = 300 and V0 = 1500 V), power deposition from the top electrode is a function of the bulk plasma properties and not independent of the bottom electrode power coupling. Power deposition as a function of phase angle φ and fundamental frequency f0 are shown in Fig. 6(a) for the top electrode (PT) and Fig. 6(b) for the bottom electrode (P0). Higher fundamental frequencies f0 on the bottom electrode produce more efficient electron heating and higher power deposition for a fixed voltage. For our conditions, electron power deposition is dominated by stochastic heating resulting from sheath expansion (or sheath speed) which is more rapid and occurs more often at higher frequencies. Increasing the fundamental frequency f0 of the voltage on the bottom electrode, therefore, leads to a significant increase in power deposition, PB. However, this relation is non-linear. A tenfold increase in frequency f0 (1–10 MHz) only results in a 1.5 (φ = 180°) to 2 (φ = 0°) increase in PB. These trends in power deposition with frequency directly translate to the corresponding trends in ne as shown in Fig. 5. Smaller phase angles have more rapidly expanding sheaths producing more electron heating, leading to an increase in PB. With constant top voltage VT, additional electron heating produced by PB which increases electron density translates to an increase in power deposition by the top electrode, PT.

FIG. 6.

Power deposition for (a) top electrode and (b) bottom electrode as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz.

FIG. 6.

Power deposition for (a) top electrode and (b) bottom electrode as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz.

Close modal
Earlier works have described the dependence of VDC on the shape of the applied waveform through the generation of the electrical asymmetry effect (EAE).24,25 Given these dependencies, VDC can be expressed as
V D C = V max + β V min 1 + β ,
(2)
where Vmax and Vmin are the maximum and minimum of the applied voltage. β is the discharge symmetry parameter given by
β = ( A p A g ) 2 n ¯ s p n ¯ s g I s g I s p ,
(3)
where the areas of the powered and ground electrodes are Ap and Ag and the ion densities near the powered and grounded surfaces are n ¯ s p and n ¯ s g. Isp and Isg are the sheath integrals, described in prior works discussing the theory of the EAE.13,24,26 This relation implies that the magnitude of VDC is largest (most negative) for φ = 0° and decreases with increasing φ (becoming more positive). The precise phases for which the extrema occur in VDC are likely sensitive to the geometry of the reactor, which determines the relative contributions of displacement and conduction current that ultimately flows to the powered electrode. The values of VDC produced by the model, shown in Fig. 7(a) as a function of phase shift φ for fundamental frequencies f0 = 1, 2, 5, and 10 MHz, generally verify this scaling. |VDC| tends to decrease with increasing φ for all frequencies (the negative VDC becoming more positive).
FIG. 7.

Plasma properties as a function of phase shift φ for different fundamental frequencies f0 = 1–10 MHz for constant voltage. (a) VDC. (b) Mean ion energy incident onto the wafer.

FIG. 7.

Plasma properties as a function of phase shift φ for different fundamental frequencies f0 = 1–10 MHz for constant voltage. (a) VDC. (b) Mean ion energy incident onto the wafer.

Close modal

An exception to this trend for VDC occurs for f0 = 1, 2 MHz, φ = 180° for which magnitude of VDC decreases slightly. These exceptions are likely due to effects such as contributions of VDC to secondary electron emission produced ionization, relative contributions of conduction current and displacement current in a geometrically complex reactor and electronegativity of the plasma which are not explicitly included in the theory leading to Eq. (2). The current that flows into the blocking capacitor through the metal substrate contains contributions of conduction current flowing through, for example, a conductive wafer and displacement current that enters through the side of the substrate that is covered by the dielectric focus ring. This displacement current originates from conduction current that charges the surface of the focus ring. However, the current that flows through the dielectric to the underlying electrode is displacement current.

The mean ion energies striking the wafer are shown in Fig. 7(b) as a function of phase shift φ for fundamental frequencies f0 = 1, 2, 5, and 10 MHz. Mean ion energy is the unweighted arithmetic average of all ions striking the wafer. The expectation is that mean ion energy should closely track the magnitude of VDC with a change in phase shift φ. This expectation is met for the highest frequency 10 MHz. At lower frequencies, beginning with 5 MHz, the correlation of mean ion energy with magnitude of VDC as a function of phase shift φ begins to break down. At 1 MHz, the mean ion energy increases with phase shift while magnitude of VDC is constant or is decreasing.

Recall that VDC results from the requirement that the current flowing to each electrode (powered and grounded) be equal over the RF cycle. In our model, currents are computed at the electrode surface. If that surface is in direct contact with the plasma, the current consists of both conduction current (charged particles flowing to the surface) and displacement current ( j D = ( ε E ) / t). If the electrode is in contact with a conductive material that is in contact with the plasma, such as the wafer, the current collected by the electrode consists of conduction current (through the wafer) and displacement current. If the electrode is buried beneath or within non-conducting dielectrics, the current the electrode collects is only displacement current. Contributions of displacement current to the bottom electrode may originate from charging of the top and sides of the focus ring. With the ratio of conduction to displacement current being functions of frequency which differs for top and bottom electrodes, there is no expectation that VDC should be independent of frequency.

Another factor that may affect VDC with frequency is electron heating. With electron heating being dominated by sheath expansion, higher bias frequencies f0 produce more electron heating adjacent to the sheath at the bottom electrode for any given voltage amplitude. With increasing f0 this increase in local as well as bulk electron density ne leads to a decrease in shielding lengths λD and a decrease in the sheath width ds at the bottom electrode. The end result is a change the ratio of surface adjacent ion densities (nsp\nsg). As described by Eqs. (2) and (3), this change directly translates to a change in the VDC.

An ion's ability to react to temporally changing electric fields in the sheath is related to the time required for the ion to cross the sheath, ΔT, compared to the RF period, 1/νRF, described by the sheath ion inertia coefficient,
S = Δ T ν R F = ν R F ( 2 d S 2 m i q V S ) 1 / 2 ,
(4)
where dS is the average sheath thickness, VS is the average sheath potential, mi is the ion mass, and q is the elementary charge. For S > 1, ions typically do not dynamically respond to changes in the electric field that occur in the sheath (high frequency, large mass, and thick sheath). Ions transiting through sheaths whose oscillation frequency produces S ≫ 1 arrive at the surface with an energy close to the average sheath potential. Although this scaling assumes a sinusoidal voltage with a single ion species, it can nevertheless serve as a general guide to understanding the frequency dependent trends in average ion energy onto the wafer shown in Fig. 7(b).

S depends on the value of the sheath width. Defining a precise criterion for location of the sheath-bulk plasma transition is an active area of research27–36 and beyond the scope of this work. In this work, the sheath edge is defined as the location at which the sheath electric field has decayed to 10−3 of its maximum value. The resulting sheath thickness is shown in Fig. 8(a) as a function of the fundamental frequency f0 with φ = 0°. Since the sheath thickness is generally inversely proportional to plasma density at the sheath edge, which increases with f0, the sheath thickness decreases with increasing frequency. S as a function of f0 for φ = 0° is shown in Fig. 8(b). Consistent with the premise above, non-dynamic ion transport through the sheath with S > 1 occurs for f0 = 10 MHz. Dynamic transport behavior where ions react to time variations of electric field in the sheath with S < 1 occurs for f0 = 1 MHz. As such, at the high end of frequencies, the ion dynamics are dominated by time averaged sheath properties, such as the DC self-bias, VDC. It is for that reason that mean ion energy as a function of phase angle for f0 = 10 MHz scales proportionally to the magnitude of VDC, as shown in Fig. 7(b).

FIG. 8.

Sheath properties as a function of fundamental frequency f0 for φ = 0° at constant voltage. (a) Sheath thickness ds and (b) sheath ion inertia coefficient S.

FIG. 8.

Sheath properties as a function of fundamental frequency f0 for φ = 0° at constant voltage. (a) Sheath thickness ds and (b) sheath ion inertia coefficient S.

Close modal

Moving to lower fundamental frequencies f0, the trend for average ion energy as a function of phase angle φ reverses. At low f0, the average ion energy increases with increasing φ. Lowering the fundamental frequency results in the sheath modulation time scales approaching a regime in which ions can dynamically react to transients in the electric fields. For example, compare the voltage waveforms shown in Fig. 2 for φ = 0° (peak) and φ = 180° (valley). Assuming dynamically reacting (positive) ions (S < 1), the maximum ion flux to the electrode occurs when the applied voltage is most negative. At φ = 0°, the applied potential is negative with respect to the plasma for most of the RF period. However, the minimum sheath potential, Vmin, has a smaller magnitude than for φ = 180°. It is this ability of the ions to (at least partially) react to applied voltage transients that results in their incident energy onto the surface to scale with the DC self-bias at high frequencies and the applied minimum potential at low frequencies.

The incident average energies of select ion species as a function of atomic mass for f0 = 10 MHz and φ = 180° are shown in Fig. 9(a). The ion energy and angular distributions corresponding to these mean energies are shown in Fig. 9(b). Due to their lower inertia, the lighter O+ ions can more rapidly react to changes in the electric field than heavier species such as CF 3 +. Consequently, the lighter ions can reach an energy closer to the maximum sheath potential drop during the small fraction of the RF period when the potential is most negative and have broader energy distributions. While the average ion energies differ by approximately 60 eV which is only about 5% of the total, these trends illustrate the relation between the average ion energy and the ion's ability to dynamically react to transient fields. That said, although the mean energies are similar, the IEADs are quantitatively different as a function of mass. These differences are highlighted by the two-decade log scale used in Fig. 9(b). As there are energy dependencies to surface reactions, there would be somewhat more sensitivity in total rates of reaction than indicated solely by mean energy.

FIG. 9.

Incident ion species as a function of mass for f0 = 10 MHz and φ = 0°. (a) Mean ion energy and (b) ion energy and angular distribution (two-decade log scale).

FIG. 9.

Incident ion species as a function of mass for f0 = 10 MHz and φ = 0°. (a) Mean ion energy and (b) ion energy and angular distribution (two-decade log scale).

Close modal

This scaling is also shown by the trends in the IEADs collected at the wafer surface shown in Fig. 10 for f0 = 1 and 10 MHz for phase shifts φ = 0°, 45°, 90°, 135°, and 180°. The IEADs represent the sum of the fluxes of all positive ions. At the lowest frequency, the increase average energy with φ is mirrored in the IEAD where the mean energy is roughly equal to −VDC − |Vmin|.

FIG. 10.

IEADs as a function of phase shift φ for fundamental frequencies (a) f0 = 1 MHz and (b) f0 = 10 MHz at constant applied voltage. Values are plotted on a two-decade log scale.

FIG. 10.

IEADs as a function of phase shift φ for fundamental frequencies (a) f0 = 1 MHz and (b) f0 = 10 MHz at constant applied voltage. Values are plotted on a two-decade log scale.

Close modal

As with the average energies onto the surface, there are significant differences in the trends for IEADs as a function of φ between the low and high fundamental frequencies. Overall, with f0 = 10 MHz, both maximum and average energies trend downwards with increasing φ while maintaining an approximately constant spread in energy.

Two distinct regimes for the behavior of average ion energy and IEAD with respect to the phase angle (and VDC) can be defined: The first is the steady state ion regime having S > 1 (large f0) in which average ion energy onto the surface is well correlated with the magnitude of VDC, with the average energy being maximum at φ = 0° and minimum at φ = 180°. The second is the dynamic ion regime having S < 1 (small f0) where average ion energy and the IEAD are dominated by the temporal response of the ions to the applied voltage waveforms—with the average ion energy being minimum at φ = 0° and maximum at φ = 180°.

To better align with industry practice where power (as opposed to voltage) is the control variable, simulations were performed in which the total power deposition was held constant for both electrodes while varying frequency and phase angle of the waveform on the bottom electrode. This was achieved by adjusting the voltage on the top electrode, VT, and bottom electrode, V0, to deliver power depositions of PT = P0 = 2000 W.

The voltage amplitudes applied to the top electrode VT are shown in Fig. 11(a) as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz. The mean electron density as a function of φ for f0 = 1–10 MHz is shown in Fig. 12. When delivering constant power, VT is a weak function of φ for a given f0. However, the VT decreases with increasing f0. For a given f0 with power delivered though the bottom electrode being held constant, the plasma density and so total impedance of the system does not significantly vary with φ. Consequently, the change in VT with φ to deliver constant power is small. However, with increasing f0, the fraction of power delivered by the bottom electrode devoted to ion acceleration decreases and that delivered to electron heating increases, which for constant power, produces an increase in electron density and decrease in impedance. A lower VT is then required to deliver constant power.

FIG. 11.

Applied voltage for constant power (2000 W) as function of phase shift φ for fundamental frequencies f0 = 1–10 MHz applied to the bottom electrode. (a) Top electrode and (b) bottom electrode.

FIG. 11.

Applied voltage for constant power (2000 W) as function of phase shift φ for fundamental frequencies f0 = 1–10 MHz applied to the bottom electrode. (a) Top electrode and (b) bottom electrode.

Close modal
FIG. 12.

Mean electron density as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz at constant applied power.

FIG. 12.

Mean electron density as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz at constant applied power.

Close modal

The voltage applied to the bottom electrode, V0, with respect to f0 and φ is shown in Fig. 11(b) for constant power deposition. Power deposition for constant voltage and f0 generally decreases with increasing φ, as shown in Fig. 5. To recoup this decrease in power to maintain constant power, V0 increases with increasing φ. The thickness of the sheath adjacent to the bottom electrode ds and the sheath ion inertia coefficient S as a function of f0 are shown in Fig. 13 for φ = 0° at constant voltage. With ds being a function of the electron density which increases with f0, the sheath thickness decreases. This reduction in ds contributes to a decrease in S. However, this effect is dominated by the changes in f0 and Vs, ultimately resulting in an increase in S with f0. This trend indicates that with constant power, a transition also occurs from the steady state ion regime to the dynamic ion regime occurs as a function of f0.

FIG. 13.

Sheath properties as a function of fundamental frequency f0 for φ = 0° at constant power. (a) Sheath thickness ds and (b) sheath ion inertia coefficient S.

FIG. 13.

Sheath properties as a function of fundamental frequency f0 for φ = 0° at constant power. (a) Sheath thickness ds and (b) sheath ion inertia coefficient S.

Close modal

The DC self-bias VDC as a function of phase angle φ for f0 = 1–10 MHz is shown in Fig. 14(a) while maintaining constant power. The trends with φ are similar those when keeping voltage constant, shown in Fig. 6(a), where the magnitude of VDC decreases (becomes more positive) with φ due to the electrical asymmetry effect. However, when holding power constant, the amplitude of V0 decreases with increasing f0 due to there being a larger proportion of power being more efficiently dissipated by electron heating. With a decrease in amplitude of V0, the magnitude of VDC decreases, becoming more positive.

FIG. 14.

Plasma parameters as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz at constant power. (a) DC self-bias and (b) mean ion energy delivered to the wafer.

FIG. 14.

Plasma parameters as a function of phase shift φ for fundamental frequencies f0 = 1–10 MHz at constant power. (a) DC self-bias and (b) mean ion energy delivered to the wafer.

Close modal

The average ion energies onto the wafer are shown in Fig. 14(b) as a function of phase shift φ for f0 = 1–10 MHz. The corresponding IEADs as a function of phase shift φ for f0 = 1 and f0 = 10 MHz are shown in Fig. 15. As when holding V0 constant, for higher frequencies the average ion energies scale with the magnitude of VDC due to the inability of the ions to dynamically react to the transients in applied voltage. For f0 = 10 and f0 = 5 MHz, this leads to a decrease in average energy onto the wafer with increasing phase angle. As with constant voltage, at low frequencies f0, the average ion energies are poorly correlated (and, in fact, anti-correlated) with VDC and instead reflect the maximum sheath potentials.

FIG. 15.

IEADs as a function of phase shift φ with constant power for fundamental frequencies (a) f0 = 1 and (b) 10 MHz. Values are plotted on a two-decade log scale.

FIG. 15.

IEADs as a function of phase shift φ with constant power for fundamental frequencies (a) f0 = 1 and (b) 10 MHz. Values are plotted on a two-decade log scale.

Close modal

As a result of the applied voltage no longer being constant, the resulting average ion energies are strong functions of phase shift φ for fundamental frequencies f0. For example, the mean energies at 1 MHz are larger than those at 10 MHz by a factor of 2 for φ = 0° and 4.5° at φ = 180°. These similarities and differences in mean ion energy between the constant voltage and constant power cases translate to the respective IEADs. At the lowest frequency (f0 = 1 MHz), the IEADs for φ = 0 and 180° are more monoenergetic, reflecting the applied waveforms. The shapes of the applied voltage for these phase shifts roughly consist of a plateau and a singular well-defined short excursion to a maximum at φ = 0° and to a minimum at φ = 180°. The intermediate cases (φ = 45°, 90°, and 135°) do not have the similar monoenergetic structures as the applied waveforms themselves do not predominantly consist of a stable voltage plateau or singular peak.

Desired anisotropic etch profiles require a balance of polymer deposition and activation energy delivered by ions and hot neutrals whose rates are largely determined but the magnitude of the reactive fluxes arriving on the surface. The generation of oxide-polymer-complexes, which enables selective removal of the SiO2 is correlated with the incident flux of the polymerizing CxFy gas phase species, which are shown as a function of φ and f0 in Fig. 16(a). The overall trends are that polymerizing fluxes are weak functions of phase angle φ while generally increasing with increasing f0 on the bottom electrode. With increasing f0, a larger proportion of bias power is dissipated by electron heating, which then produces a larger rate of dissociation of the CF4 feedstock gas. With increasing phase angle, the sheath is collapsed for a greater fraction of the cycle, thereby moving the effective (time average) sheath edge closer to the wafer. This shift in sheath edge places radical production closer to the wafer, and so increases fluxes.

FIG. 16.

Fluxes to the wafer as a function of phase shift φ for fundamental frequencies f0 = 1 to 10 MHz while keeping power constant: (a) total polymerizing flux, (b) O-atom flux, (c) total ion flux, and (d) ratio of polymerizing flux to ion flux.

FIG. 16.

Fluxes to the wafer as a function of phase shift φ for fundamental frequencies f0 = 1 to 10 MHz while keeping power constant: (a) total polymerizing flux, (b) O-atom flux, (c) total ion flux, and (d) ratio of polymerizing flux to ion flux.

Close modal

The thickness of the polymer layer by deposition is balanced by chemical, isotropic etching by O radicals and anisotropic sputtering by directional ions. The time and spatial average of fluxes to the wafer of atomic oxygen, O, the most prevalent oxygen radical are shown in Fig. 16(b). The general trends reflect those of the polymerizing fluxes [Fig. 16(a)] that increase with φ and f0.

The magnitude of the ion flux, shown in Fig. 16(c), has a first order effect on etch rates and feature quality. Applied electric fields directly influence charged particle transport, whereas electron impact dissociation, excitation, and ionization are at least one step removed as these phenomena occur as a result of electron collisions following their acceleration by electric fields. The dominant mechanism for ion power deposition is through sheath acceleration, which assuming a collision-less sheath, is proportional to the product of ion flux and incident ion energy. Assuming a constant fraction of power deposition by ion acceleration, a decrease in incident ion energy would necessitate an increase in ion flux to maintain the desired power. This is the trend for f0 = 5 and 10 MHz in which the trend of ion-flux with phase angle is opposite that of the ion energy. This explanation fails to capture the low frequency behavior in ion flux where an increasing fraction of the constant power deposition is due to electron heating.

To characterize the consequences of reactant fluxes and IEADs on SiO2 etch properties, profile simulations were performed for f0 = 1 and 10 MHz for phase angles φ = 0°–180° while maintaining constant power. The resulting features for f0 = 1 MHz are shown in Fig. 17.

FIG. 17.

Predictions for etch profiles in SiO2 at constant power with fundamental frequency f0 = 1 MHz and varying phase angles φ.

FIG. 17.

Predictions for etch profiles in SiO2 at constant power with fundamental frequency f0 = 1 MHz and varying phase angles φ.

Close modal

For constant processing time, etching through the entire 3000 nm thick SiO2 layer only occurred for φ = 180°, while for φ = 0° the final etch depth is 1600 nm. The total relative etch rates are 0.53, 0.6, 0.78, 0.93, and 1.0 for φ = 0°, 45°, 90°, 135°, and 180°, respectively. This trend is directly correlated with the trend in incident ion energy. Higher ion energies generally more rapidly remove SiO2 by direct or chemically enhanced sputtering and retain their ability to do so after losing energy to grazing sidewall collisions. The profiles produced by IEADs at higher φ have more desirable characteristics such as straighter sidewalls and less overall bowing. This benefit is a direct consequence of the narrower angular distribution of the incident ions. The SiO2 etch mechanism contains a SiO2–polymer complex that requires fluorocarbon radical fluxes as reactants. Since these fluxes have a small increase with φ [Fig. 16(a)], the increased availability of reactants could also play a role in the increased etch rate if the etch progression is flux limited as is often the case in HAR features.

The results of the feature etching for f0 = 10 MHz are shown in Fig. 18. The overall etch rates are similar to those at f0 =1 MHz while the incident ion energies are significantly lower overall. The trend in etch depth with phase angle is the opposite to that for f0 = 1 MHz. With f0 = 10 MHz, etch rates mildly decrease with increasing phase angle φ while increasing for f0 = 1 MHz. These trends most directly follow from both the average ion energies and maximum ion energies trending higher with increasing phase angle φ at f0 = 1 MHz while decreasing at f0 = 10 MHz. A secondary effect is that ion fluxes are nearly constant with increasing phase angle at f0 = 10 MHz while increasing at f0 = 1 MHz. With chemical and physical sputtering rates depending on ion energy as ε1/2, large increases in ion energy are required for significant increases in etch rates. That said, the likelihood for specular scattering from side walls increases with increasing ion energy, and so more energy is retained deeper into the feature upon grazing collisions with sidewalls. On the other hand, for otherwise constant, non-rate limiting conditions, etch rates increase linearly with increases in ion flux.

FIG. 18.

Predictions for etch profiles in SiO2 at constant power with fundamental frequency f0 = 10 MHz and varying phase angles φ.

FIG. 18.

Predictions for etch profiles in SiO2 at constant power with fundamental frequency f0 = 10 MHz and varying phase angles φ.

Close modal

While certainly an important parameter, ion energy is not the sole determining factor for etch rate. The etch process is based on a sensitive balance of surface passivation, activation, and removal by fluxes of neutrals, ions, and hot neutrals (generated by ions neutralizing during surface collisions). For each fundamental frequency, the ratio of these fluxes as function of φ is relatively constant, as shown in Fig. 16(d). Although ion energy and fluxes determine overall rate of etching, the shape of the feature (e.g., sidewall slope, bowing) depends on relative rates of passivation by deposition, etching, and sputtering. For example, for f0 = 10 MHz, despite the higher etch rate with increasing phase angle, the features generally have less bowing, an effect that may be attributable to a larger ratio of polymerizing flux to ion flux. Large fluxes of passivating species typically produce more tapered features.

Coupled reactor and feature scale simulations were performed to investigate the consequences of the fundamental driving frequency f0 on the relation between DC self-bias VDC, incident ion energy onto the wafer, and reactive fluxes in dielectric etch processes using tailored voltage waveforms as a power source. For a set of waveforms based on consecutive harmonics for which the relative phase angle φ was varied from 0° to 180°, gas phase simulations were performed using fundamental frequencies f0 = 1, 2, 5, and 10 MHz. While plasma conditions were found to differ when holding either voltage or power constant, the trends in VDC were qualitatively similar for the two scenarios. The magnitude of VDC decreases (a negative bias becoming more positive) with increasing φ for all f0 at constant voltage as well as power. Due to ion inertia, in the high frequency regime ion fluxes to the substrate are dominated by time average quantities such as VDC. At low f0, the ions are able to react to transient characteristics in the sheath such as local extremes in the sheath potential directly produced by the applied tailored voltage waveforms.

The consequences of inertial effects on IEADs as a function of sinusoidal bias frequency are well known. The thick sheath regime corresponding to high frequency and large ion mass produces a single peaked IEAD. The thick sheath regime corresponding to low frequency and small mass produces a double peaked (bimodal) IEAD. The extension of these dependencies to VWT power sources is not straightforward due to the intrinsically more complex sheath structure and harmonic content of the applied voltages. The majority of sheath theories are based on applying a single sinusoidal voltage. VWT applies several discrete frequencies that induce additional nonlinearities. The extension to square-wave pulsing with even more Fourier derived frequencies present additional theoretical challenges.

The results of this study suggest that scaling of VDC produced by processes akin to the electrical asymmetry, a common goal sought when using voltage waveform tailoring, is particularly sensitive to the frequency regime of f0. Low values of f0 can result in significantly different trends of incident ion energy as a function of phase angle compared to high values of f0. Incident ion energy is closely correlated with the VDC self-bias at high values of f0, maximum at φ = 0° and minimum at φ = 180°. For low values of f0, this correlation dissipates, and incident ion energy is instead dominated by the sheath dynamics of the applied voltage waveforms, maximum at φ = 180° and minimum at φ = 0°.

The trends in ion energy (and IEADs) as a function of f0 and φ directly impact the HAR etch process. Although the observations and conclusions made in this work are highly dependent on power deposition, plasma density, chemical composition, and geometry, these observations also open additional avenues for process control. With three-dimensional structures and atomic layer resolution already dominating industrial plasma etching processes, additional control strategies are required to achieve the desired critical dimensions. With frequency agile power supplies becoming more available, the combination of using VWT while varying fundamental frequency and phase angle provide new control opportunities.

The authors are not aware of the availability of experimental data for a 1-to-1 comparison to the results discussed here. The modeling platform that has been used in this investigation is accurate in the sense that results using the model have been validated against experiments performed in many types of reactors and chemistries. The degree of accuracy for any specific investigation is difficult to assess when that investigation is in a parameter space that has not yet been, or not been fully, experimentally investigated. Modeling is extremely valuable in attempting to reproduce experimental results to confirm theories, tasks that are typically performed in simpler geometries and less complex gas mixtures. This practice reduces the uncertainties associated with incomplete reaction mechanisms and produced by geometrical effects, and so enables one to focus on the theory. Modeling was used here to investigate a parameter space that has not been fully experimentally characterized for conditions that are close to those used in applications to help guide those applications. In doing so, the expected precision of the computational investigation is likely less than more focused investigations. The value of investigations of this type is in developing scaling laws that are generally applicable to guide technology development.

This work was supported by Samsung Electronics Ltd and Tokyo Electronics America. These results were also based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award No. DE-SC0020232 and supported by the National Science Foundation (No. PHY-2009219).

The authors have no conflicts to disclose.

Florian Krüger: Conceptualization (lead); Formal analysis (lead); Investigation (lead); Methodology (lead); Writing – original draft (lead). Hyunjae Lee: Resources (equal); Supervision (equal); Writing – review & editing (equal). Sang Ki Nam: Resources (equal); Supervision (equal); Writing – review & editing (equal). Mark J. Kushner: Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

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

1.
V. M.
Donnelly
and
A.
Kornblit
,
J. Vac. Sci. Technol., A
31
,
050825
(
2013
).
2.
Y.
Li
,
IEEE Solid-State Circuits Mag.
12
,
56
(
2020
).
3.
S.-J.
Chung
,
P.
Luan
,
M.
Park
,
A.
Metz
, and
G. S.
Oehrlein
,
J. Vac. Sci. Technol., B
41
,
062201
(
2023
).
4.
S.
Rauf
and
A.
Balakrishna
,
J. Vac. Sci. Technol., A
35
,
021308
(
2017
).
5.
B.
Wu
,
A.
Kumar
, and
S.
Pamarthy
,
J. Appl. Phys.
108
,
051101
(
2010
).
6.
G. A.
Skarphedinsson
and
J. T.
Gudmundsson
,
Plasma Sources Sci. Technol.
29
,
084004
(
2020
).
7.
S.
Brandt
,
B.
Berger
,
E.
Schüngel
,
I.
Korolov
,
A.
Derzsi
,
B.
Bruneau
,
E.
Johnson
,
T.
Lafleur
,
D.
O'Connell
,
M.
Koepke
,
T.
Gans
,
J.-P.
Booth
,
Z.
Donkó
, and
J.
Schulze
,
Plasma Sources Sci. Technol.
25
,
045015
(
2016
).
8.
P.
Hartmann
,
L.
Wang
,
K.
Nösges
,
B.
Berger
,
S.
Wilczek
,
R. P.
Brinkmann
,
T.
Mussenbrock
,
Z.
Juhasz
,
Z.
Donkó
, and
A.
Derzsi
,
J. Phys. D
54
,
255202
(
2021
).
9.
Z.
Donkó
,
A.
Derzsi
,
M.
Vass
,
J.
Schulze
,
E.
Schuengel
, and
S.
Hamaguchi
,
Plasma Sources Sci. Technol.
27
,
104008
(
2018
).
10.
T.
Lafleur
,
Plasma Sources Sci. Technol.
25
,
013001
(
2015
).
11.
J.
Wang
,
S.
Dine
,
J.-P.
Booth
, and
E. V.
Johnson
,
J. Vac. Sci. Technol. A
37
,
021303
(
2019
).
12.
F.
Schmidt
,
J.
Schulze
,
E.
Johnson
,
J.-P.
Booth
,
D.
Keil
,
D. M.
French
,
J.
Trieschmann
, and
T.
Mussenbrock
,
Plasma Sources Sci. Technol.
27
,
095012
(
2018
).
13.
J.
Schulze
,
E.
Schüngel
,
Z.
Donkó
, and
U.
Czarnetzki
,
Plasma Sources Sci. Technol.
20
,
015017
(
2011
).
14.
P.
Saikia
,
H.
Bhuyan
,
M.
Escalona
,
M.
Favre
,
B.
Bora
,
M.
Kakati
,
E.
Wyndham
,
R. S.
Rawat
, and
J.
Schulze
,
J. Appl. Phys.
123
,
183303
(
2018
).
15.
C.
Qu
,
Y.
Sakiyama
,
P.
Agarwal
, and
M. J.
Kushner
,
J. Vac. Sci. Technol., A
39
,
052403
(
2021
).
16.
S. J.
Lanham
,
J.
Polito
,
Z.
Xiong
,
U. R.
Kortshagen
, and
M. J.
Kushner
,
J. Appl. Phys.
132
,
073301
(
2022
).
17.
S.
Huang
,
S.
Shim
,
S. K.
Nam
, and
M. J.
Kushner
,
J. Vac. Sci. Technol., A
38
,
023001
(
2020
).
19.
H. K.
Kim
,
T. S.
Kim
,
J.
Lee
, and
S. K.
Jo
,
Phys. Rev. B
76
,
165434
(
2007
).
20.
A. V.
Phelps
and
Z. L.
Petrovic
,
Plasma Source Sci. Technol.
8
,
R21
(
1999
).
21.
S.
Huang
,
C.
Huard
,
S.
Shim
,
S. K.
Nam
,
I.-C.
Song
,
S.
Lu
, and
M. J.
Kushner
,
J. Vac. Sci. Technol., A
37
,
031304
(
2019
).
22.
C. M.
Huard
,
S.
Sriraman
,
A.
Paterson
, and
M. J.
Kushner
,
J. Vac. Sci. Technol., A
36
,
06B101
(
2018
).
23.
J.
Schulze
,
Z.
Donkó
,
B. G.
Heil
,
D.
Luggenhölscher
,
T.
Mussenbrock
,
R. P.
Brinkmann
, and
U.
Czarnetzki
,
J. Phys. D
41
,
105214
(
2008
).
24.
U.
Czarnetzki
,
J.
Schulze
,
E.
Schüngel
, and
Z.
Donkó
,
Plasma Sources Sci. Technol.
20
,
024010
(
2011
).
25.
S. J.
Doyle
,
A. R.
Gibson
,
R. W.
Boswell
,
C.
Charles
, and
J. P.
Dedrick
,
Plasma Sources Sci. Technol.
29
,
124002
(
2020
).
26.
B. G.
Heil
,
U.
Czarnetzki
,
R. P.
Brinkmann
, and
T.
Mussenbrock
,
J. Phys. D
41
,
165202
(
2008
).
27.
L. P.
Beving
,
M. M.
Hopkins
, and
S. D.
Baalrud
,
Plasma Sources Sci. Technol.
31
,
084009
(
2022
).
28.
B. G.
Heil
,
R. P.
Brinkmann
, and
U.
Czarnetzki
,
J. Phys. D
41
,
225208
(
2008
).
29.
T.
Shirafuji
and
K.
Denpoh
,
Jpn. J. Appl. Phys., Part 1
57
,
06JG02
(
2018
).
30.
B.
Mancinelli
,
L.
Prevosto
,
J. C.
Chamorro
,
F. O.
Minotti
, and
H.
Kelly
,
Plasma Chem. Plasma Process.
38
,
147
(
2018
).
31.
O.
Murillo
,
A. S.
Mustafaev
, and
V. S.
Sukhomlinov
,
Tech. Phys.
64
,
1308
(
2019
).
32.
M. S.
Benilov
and
N. A.
Almeida
,
Phys. Plasmas
26
,
123505
(
2019
).
33.
S. D.
Baalrud
,
B.
Scheiner
,
B. T.
Yee
,
M. M.
Hopkins
, and
E.
Barnat
,
Plasma Sources Sci. Technol.
29
,
053001
(
2020
).
34.
35.
36.
M.
Klich
,
J.
Löwer
,
S.
Wilczek
,
T.
Mussenbrock
, and
R. P.
Brinkmann
,
Plasma Sources Sci. Technol.
31
,
045014
(
2022
).