Azimuthal ion movement in HiPIMS plasmas—part I: velocity distribution function

A cylindrical vacuum chamber with a diameter of 25 cm and a height of 40 cm was used for the experiment. It was pumped to a base pressure of $4 \times 10^{-6}$ Pa. Argon was used as working gas at a pressure of 0.5 Pa. A planar 2" magnetron (Thin Film Consulting IX2U) in combination with a TRUMPF Hüttinger power supply (TruPlasma Highpulse 4002) was used to drive the plasma discharges.

The discharge was monitored by current and voltage measurements with commercial probes (Tektronix TCP A400, Tektronix P6015A) attached to the connection cable between the power supply and the magnetron assembly. Discharge conditions were selected to be the same as in earlier publications [20, 36]. The applied voltage was −590 V with a repetition frequency of 40 Hz and a pulse length of 100 µs. Using titanium targets, these values result in peak currents of 50 A, peak target-area-normalized current densities of 2.5 A cm⁻² and peak power densities of 1.1 kW cm⁻². The corresponding voltage and current waveforms can be found in a previous publication [36].

The setup for the high-resolution optical emission spectroscopy was adapted from [20] and is shown in figure 1(a). The plasma is observed parallel to the target surface. A convex lens (f = 150 mm) is used to collect the emitted light and couple it into an optical fiber (Ø = 800 µm). The distance between lens and fiber is adjusted to limit the field of view of the system to a narrow cone (figure 1(b)). The focal spot has a diameter of approximately 2 mm with the focal plane adjusted to the center of the target. The whole lens system is mounted on a movable stage and can be moved along the magnetron axis or parallel to the target surface (z and x direction). As illustrated in figure 1(b), we define the z-axis of our coordinate system in target normal or axial direction and the x and y-axis parallel to the target surface with the origin in the center of the target. Additionally, the coordinate ϕ is used to describe the azimuthal ion movement. It points in the same direction as the $\vec{E} \times \vec{B}$ drift.

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Measurements of the emission lines were performed with an intensified CCD-Camera (Andor iStar DH320T-25U-A3) attached to a 2 m plane grating spectrograph (Zeiss Jena PGS 2, 1300 lines/mm grating). All measurements were performed at the end of the discharge pulse by triggering the camera with a delay of $\Delta t = 90\, \mu \text{s}$ to the plasma ignition. The gate width was set to 10 µs and the data were accumulated over 2000 plasma pulses, with the exception of the x-scan presented in figure 3(b), where the last 15 µs of the pulse were measured and 500 accumulations were used, instead. By operating the spectrograph in the third diffraction order, an spectral resolution of 1.5 pm pixel-to-pixel at the camera chip was achieved. To enable calibration of the wavelength axis for the measured spectra the emission from a hollow cathode lamp (HCL, Cathodeon 3UNX Ti) was measured simultaneously with the plasma emission, as indicated in figure 1(a). Details about the used emission lines including the involved energy levels are displayed in table 1.

Table 1. Energy levels and electron configurations of the selected optical emission lines [37, 38] collected from the NIST atomic spectra database [39]. The electron configuration is given in the notation first proposed by Russel et al [40].

Species	Ti II	Ar II	Ti I
Wavelength (nm)	453.396	472.687	453.324
Upper level
Energy (eV)	3.97	19.76	3.58
e−-config.	$3d^2(^3F)4p$	$3s^23p^4(^3P)4p$	$3d^3(^4F)4p$
Lower level
Energy (eV)	1.24	17.14	0.85
e−-config.	$3d^3$	$3s^23p^4(^3P)4s$	$3d^3(^4F)4s$

The determination of the VDF from the emission lines was performed as described by [36]. The method is based on the analysis of the two dominant line broadening mechanisms, Doppler broadening and instrumental broadening. As a first step, a Wiener deconvolution is used to remove the contribution of instrumental broadening to obtain an emission line profile only affected by Doppler broadening. Afterwards, the wavelength axis of the spectra is transformed into a velocity axis using the relation $v = c(\frac{\lambda}{\lambda_0} - 1)$, where λ₀ is the wavelength of the emission line in an unshifted state measured from the emission by the HCL. An exemplary measurement is shown in figure 2, which shows the considered Ti I and Ti II lines at 453.324 nm and 453.396 nm, respectively. The light of a HCL is simultaneously measured to acquire an unshifted wavelength reference for Ti I. For Ti II, the lamp emission is not bright enough and both of the neutral lines visible in the lamp spectrum in figure 2 are used as a reference, utilizing the tabulated wavelength difference between these lines and the considered Ti II line, according to NIST [37, 39]. Further details can be found in previous publications [20, 36].

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Probe measurements were performed above the racetrack position, in target distances of 6.3, 8.0 and 9.7 mm. The probe setup [3] and results [41] are discussed in great detail in recent publications. Here, we only use the electron density and plasma potential obtained from those measurements to estimate the physical background and the corresponding forces acting on the ions.

The dependence of the average velocity on the distance to the target surface is shown in figure 4(a). Considering argon ions (Ar II) first, we initially observe a steep increase with increasing target distance from about $v_{\varphi \text{ArII}} = 0.8$ km s⁻¹ at z = 1.5 mm to a maximum of $v_{\varphi \text{ArII}} = 1.55$ km s⁻¹ around z = 7 mm. The gap in the measurement data around z = 5 mm is caused by the anode cover, blocking the field of view of the optical measurement system, as explained in a recent publication [42]. For higher values of z, $v_{\varphi \text{ArII}}$ begins to decrease with target distance and vanish at z = 23 mm. A similar trend is observed for Ti II, with a smaller maximum velocity of $v_{\varphi \text{TiII}} = 1.25$ km s⁻¹ peaking at a slightly larger target distance of z = 10 mm.

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Qualitatively, the observed trends in azimuthal ion velocity with varying target distance can be understood by considering the trajectories of the ions. For titanium, particles are created by sputtering at the target surface and are then ionized very close to the target, at z < 1 mm, as we recently reported [41, 43]. Particles entering the plasma from the target exhibit an initial mean azimuthal velocity of $v_{\varphi} = 0$ and are subsequently accelerated inside the plasma. Thus, v_ϕ increases with target distance for Ti II as particles travel through the plasma and are continuously accelerated in the azimuthal direction. The v_ϕ decrease at larger target distances z > 10 mm can be explained by the lack of any additional acceleration force at these positions, independent of which physical process is actually causing the acceleration: waves are expected to be much weaker at this position [3, 30, 32] and the electron density much lower, leading to less momentum transfer to the ions. Instead, the ions are only slowed down by collisions with the background gas, leading to smaller azimuthal velocities. On top of that, only the fastest ions can overcome the electric field and reach positions with z > 10 mm. Because of their large velocity, such ions have crossed the dense plasma region close to the target very quickly, leaving them not much time to be accelerated in azimuthal direction. This effect will be explored in more detail in section 3.3.

The larger maximum azimuthal velocity for argon ions compared to titanium ions can simply be explained by their smaller mass: the effective force acting on the ions is independent of the ion mass according to the reported explanations for the azimuthal ion movement found in the literature. As such, the ion acceleration is expected to scale with the ion mass m as m⁻¹. For equal residence times within the plasma, this would lead to a difference in the maximum velocity of $m_\mathrm{Ti}/m_\mathrm{Ar} = 1.2$, which accounts for almost all of the observed differences in azimuthal velocity.

On top of the difference in maximum azimuthal velocity, figure 4(a) also shows that the position of peak velocity is different for the two ion species: for titanium ions, the velocity peaks at z = 10 mm, whereas the peak position is at about z = 7 mm for argon ions. This shift can likely be explained by the difference in location where ionization occurs for the different ion species. As we recently reported [41], titanium neutrals get ionized in the direct vicinity to the target surface, z < 1 mm. As such, titanium ions necessarily maintain a positive flow velocity (away from the target), which means that titanium ions located further away from the target have spent a longer time within the plasma to reach this position. Since the azimuthal ion velocity increases the longer the ions spend within the plasma (as they are continuously accelerated), this positive flow velocity in z direction will shift the maximum azimuthal velocity to higher z-positions. In contrast, argon is ionized everywhere within (and even outside) the magnetic trap region. As the argon ions are attracted towards the target by the electric field, they assume a negative flow velocity (towards the target) which shifts the maximum in the azimuthal velocity towards smaller z-positions. The smaller argon ion velocity closer to the target surface, z < 7 mm might partly be caused by Coulomb collisions with the slower titanium ions and partly by mixing with argon ions created from neutrals that have outgassed from the target surface due to the working gas recycling [44].

For titanium neutrals, figure 4(a) reveals a maximum azimuthal velocity of 0.45 km s⁻¹, located somewhere around z = 10 mm. Data for z > 13 mm could not be obtained, since the emission line used for the measurement was disturbed by titanium ion emission, which increases in relative intensity with the target distance. The observation that even neutral species have a considerable velocity in azimuthal direction is at first surprising, since neither of the two explanations proposed in the literature for the azimuthal acceleration—momentum transfer from the electrons or the electric field fluctuations caused by the spokes—applies to neutrals. We propose that the movement of neutrals is due to resonant charge exchange collisions with the titanium ions. Since titanium ions are expected to have a large density and the cross section for resonant charge exchange is very large for titanium ($\sigma_\mathrm{cx} \approx 2 \times 10^{-18}$ m² [45]), the titanium neutrals are being dragged along with the azimuthal movement of ions. Assuming a titanium ion density of $n_{\mathrm{Ti}^+} = 5 \times 10^{19}$ m⁻³, the mean free path for resonant charge exchange is only $\lambda = (n_{\mathrm{Ti}^+} \sigma_\mathrm{cx})^{-1} = 10$ mm, demonstrating that our hypothesis is reasonable. This explanation is also in good agreement with the previously observed close coupling of titanium neutral and ion VDF in target normal direction [36].

For argon neutrals, no azimuthal drift could be observed with our setup. However, previous work by Kanitz et al did reveal a mean azimuthal velocity of about 30 m s⁻¹ for argon metastable atoms [46]. This much lower azimuthal velocity can be explained by the smaller cross-section for resonant charge exchange for argon and the lower argon ion density in the target vicinity, leading to much less efficient momentum transfer from ions to neutrals.

Figure 4(b), shows the width (full width at half maximum—FWHM) of the VDF, as a measure of the average energy or effective temperature of the species. For titanium ions, the width of the VDF was already discussed in a recent publication [20]. There, we explained that titanium ions start their life as highly energetic sputtered particles following a Thompson energy distribution. The Thompson distribution is rather narrow (in terms of FWHM), but features a strongly populated high-energy tail. As particles move through the plasma they undergo Coulomb collisions with each other, leading to the relaxation of the VDF towards a Maxwell distribution. At the same average energy, the Maxwell distribution has a much larger FWHM, which is why the FWHM in figure 4(b) increases with target distance until about z = 8 mm after which cooling by collisions with the background gas causes the VDF to become more narrow again. The maximum width observed here for Ti II is around 11 km s⁻¹, which would correspond to a temperature of about 10 eV in case of a fully relaxed distribution.

For argon ions, we generally find a much narrower VDF, with FWHMs between 5 km s⁻¹ and 8 km s⁻¹. The reason for this smaller VDF width, which corresponds to a lower average energy, is that argon ions are created from argon neutrals, which are known to remain comparatively cold during the discharge pulse [46, 47]. Even argon created from working gas recycling is expected to only have a temperature on the order of 1000 K which corresponds to a FWHM of only about 1 km s⁻¹ [44, 48]. As such, the newly ionized argon particles start out cold and are then heated up by Ohmic heating and by collisions with the energetic titanium ions, which leads them to acquire an effective temperature somewhere between the temperature of the cold argon neutrals and of the highly energetic titanium ions. The maximum VDF width observed for the argon ions (8 km s⁻¹) corresponds to a temperature of 4.8 eV.

For titanium neutrals, the VDF has an initial width of about 8.5 km s⁻¹, corresponding to an unaltered Thompson distribution. For larger target distances, the VDF becomes slightly more narrow, presumably due to collisions with the working gas.

Following the qualitative description of the observed azimuthal velocities, we will now attempt to find a quantitative description. To this end, forces in both axial (z) as well as azimuthal (ϕ) direction need to be considered. The forces in z direction determine the residence time of the particles within each volume element, which determines how much acceleration in ϕ direction the passing species can accumulate.

3.2.1. Force in z direction and electric field.

The movement of ions in z-direction is mainly determined by the electric field in the magnetic trap region of the plasma, caused by the limited mobility of electrons across the magnetic field lines [49]. As such, we need to find an estimation of the electric field configuration to describe the ion movement in this direction.

The electric field $\vec{E}$ is derived from the topology of the magnetic field $\vec{B}$ following a simple physical argument: due to the high mobility of electrons parallel to the magnetic field lines, any potential differences inside the magnetic trap region can only occur perpendicular to those. Therefore, the magnetic flux coordinates Ψ introduced by [50] can be used to construct the topology of the electric potential Φ inside the magnetic trap region. Since this approach can only produce the topology of the plasma potential but not its absolute values, a specific scaling has to be assumed. This is achieved, by adjusting the electric potential to be consistent with probe measurements, which were performed at distances of 6.3, 8.0 and 9.7 mm. The magnetic field configuration, obtained from Hall-probe measurements following the method of [15], can be found in a previous publication [20].

Figure 5(a) shows the reconstructed plasma potential assuming $\Phi \propto \Psi$. The arrows in the figure indicate the direction of the electric field, perpendicular to the magnetic field lines. This plasma potential topology is in good agreement with measurements of Rauch et al and Mishra et al [13, 51].

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Figure 5(b) shows the potential above the racetrack position (r = 13.5 mm) as well as the electric field in axial direction E_z calculated from the potential. This electric field can now be used to model the particle movement in z direction as:

$$\begin{equation} \frac{\mathrm{d} v_z}{\mathrm{d} t} = \frac{e E_z}{m}. \end{equation} \tag{ 1 }$$

3.2.2. Forces in ϕ direction and electron density.

During their movement in z-direction, the particles are accelerated in azimuthal direction. In contrast to the prior work from the literature [21–23], we do not consider a two stream instability or additional azimuthal electric field, but only the momentum transfer from electrons to ions via Coulomb collisions.

The drag force acting on the ions $F_\mathrm{drag}$ can be calculated as

$$\begin{equation} F_\mathrm{drag} = \eta_{\perp} e^2 n_e v_e \end{equation} \tag{ 2 }$$

with the electron density n_e , the electron drift velocity in azimuthal direction v_e and the Spitzer resistivity for the direction perpendicular to the magnetic field lines $\eta_{\perp}$. For a highly ionized plasma, $\eta_{\perp}$ can be calculated as [52]:

$$\begin{equation} \eta_{\perp} = \frac{2 \pi e^2 \sqrt{m_e} }{\left(4 \pi \epsilon_0\right)^2 \left(k_B T_e\right)^{3/2}} \ln \Lambda \end{equation} \tag{ 3 }$$

with the electron mass m_e , the electron temperature $T_e = 4.5$ eV and $\Lambda = 12 \pi n_e \lambda_D^3$ depending on the Debye length λ_D .

The electron drift velocity v_e consists of the $\vec{E} \times \vec{B}$ drift, the curvature drift and the diamagnetic drift [15, 52]:

$$\begin{align} {v}_{E\times B} & = \frac{\vec{E} \times \vec{B}}{B^2} \end{align} \tag{ 4 }$$

$$\begin{align} {v}_{c} & = -\frac{v_{||}^2}{\omega_c} \vec{b} \times \left(\vec{b}\cdot \nabla\right)\vec{b} \end{align} \tag{ 5 }$$

$$\begin{align} {v}_\mathrm{dia} & = T_e \frac{\nabla p \times \vec{B}}{e n_e B^2} \end{align} \tag{ 6 }$$

with the electron cyclotron frequency $\omega_c = eB/m_e$, the unit vector $\vec{b} = \vec{B}/B$, and the electron velocities $v_{||}$ parallel and $v_\perp$ perpendicular to the magnetic field. These drift velocities, together with equations (2) and (3) can be used to calculate the drag force acting on the ions as they travel through the plasma, leading to acceleration in the azimuthal direction. However, an estimation for the electron density is required for the diamagnetic drift as well as the momentum transfer from electrons to ions.

3.2.2.1. Electron density.

The spatial dependence of the electron density was estimated from the discharge current, from Langmuir probe measurements as well as from optical measurements. The maximum of the electron density n_e is expected to be located at the pre-sheath edge close to the target surface. This density can be estimated from the measured discharge current I and the velocity v_s with which the ions enter the sheath as:

$$\begin{equation} n_e \approx \frac{2 I}{0.61 e v_s A} \end{equation} \tag{ 7 }$$

with the target surface area A and the factor of two accounting for the difference between the average current density across the target surface and the larger local current density above the racetrack [2, 43] and neglecting multiply charged ions as well as secondary electrons. According to the Bohm criterion, v_s will either be the ion sound speed $v_{s} = \sqrt{k_B T_e/M}$, or a larger value, which might be the case in HiPIMS plasmas due to the strong acceleration ions experience in the electric field present in the magnetic trap. For the former case, we find $v_{s} \approx 3$ km s⁻¹, using $T_\mathrm{e} = 4$ eV [41] and assuming a mix of titanium and argon ions with an average mass of M = 44 u. For the other case, v_s is instead determined by the local electric field strength E and the acceleration length L as $v_{s} = \sqrt{\frac{2 E L e}{M}}$. Since ionization occurs very close to the target surface under these conditions [41], we estimate L = 0.5 mm. Together with E = 4 KV m⁻¹ we again obtain $v_{s} \approx 3$ km s⁻¹. From this, we estimate a maximum density of $n_e \approx 1.8 \times 10^{20}$ m⁻³.

The density was also measured using a Langmuir probe at distances of 6.3, 8.0 and 9.7 mm. The spatial distribution of the electron density in z direction was further assumed to roughly follow the square root of the titanium ion emission, since this emission depends on the product of electron density n_e and titanium ion density $n_{Ti}^+$. This relationship requires singly charged titanium ions to be the dominant ion species and a constant electron temperature, which is not the case. But the relationship can still be useful as an indication of the expected shape of electron density distribution. The titanium ion emission was obtained using Abel-inverted optical imaging, using a recently described setup [41, 43]. Based on these three pieces of information we approximate the spatial electron density profile as:

$$\begin{align} n_\mathrm{e}\left(z\right) & = \frac{2.47 \times 10^{20}\, \mathrm{m}^{-3}}{\exp\left(L_1/z\right)+1} \left( \exp\left(-z/L_2\right)\right. \nonumber\\ & \quad \left.+ \exp\left(-z/L_3\right) \right) + 2.5 \times 10^{19}\,\mathrm{m}^{-3} ~ \end{align} \tag{ 8 }$$

with $L_1 = 0.1$ mm, $L_2 = 0.8$ mm and $L_3 = 4.5$ mm. This equation is purely meant to fit the probe data, the density from the sheath edge and the square root of the Ti II emission as close as possible but has no physical meaning or origin beyond that.

The resulting electron density is shown figure 6, together with the probe measurements and the Abel inverted titanium ion emission, for comparison.

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Based on the proposed electric field $\vec{E}(z)$, magnetic field $\vec{B}(z)$, and electron density $n_e(z)$, the different electron drift velocities as well as their sum can be calculated, as shown in figure 7(a). Diamagnetic and $\vec{E} \times \vec{B}$ drift are almost constant along z, which is a consequence of assuming the gradients for electron density and electric field to be similar as the gradients in the magnetic field configuration. For the diamagnetic drift and $\vec{E} \times \vec{B}$ drift, we find drift velocities of about 15 km s⁻¹ and 26 km s⁻¹, respectively. In contrast, the curvature drift velocity increases from about 20 km s⁻¹ close to the target to about 68 km s⁻¹ at 10 mm and then begins to decrease. All drift velocities are set to zero for z > 12.3 mm, because the gyroradius r_L of the electrons becomes larger than half the gradient length scale of the magnetic field $B/\nabla B$, which we use as criterion for electron magnetization [15]. One can state that all drift mechanisms contribute similarly to the overall azimuthal electron drift velocity yielding values in the order of 100 km s⁻¹, as expected [16, 53].

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From the calculated drift velocities, we can now determine the azimuthal force acting on the ions, using equation (2). Figure 7(b) shows this drag force and the individual contributions from the different drift velocities as a function of target distance. The drag force peaks close to the target surface due of the large electron density, but then decreases at larger target distances.

The model described above is solved using a test-particle Monte Carlo simulation with one spatial (z) and two velocity dimensions (v_z and v_ϕ ). This method was first used by [54] and is comparable to a PIC simulation in which the fields are specified a priori [55], i.e. without calculating them self-consistently. Thus, all obtained results are determined by the forces estimated above, which act as an input for the simulation. The simulation considers an ensemble of 10⁷ particles that propagate in space according to their velocity and are accelerate according to the forces within small time steps $\Delta t = 75$ ns. Titanium ions are considered as the test particles and are introduced at z = 0, corresponding to ionization very close to the target surface [41].

Particles are initialized with a Thompson distribution and are then accelerated by the electric field in z direction and by the electron drag force in ϕ direction. Particles are removed from the simulation if they either leave the simulation volume by moving beyond z = 40 mm, or if they return to the target. In both cases, a new test particle is created at z = 0, to keep the total amount of particles constant. The simulation is performed over a time of 60 µs, ensuring full convergence to constant ion density and azimuthal ion velocity distributions. Collisions are neglected in the simulation since the densities of the main collision partners are unknown. However, we expect the effect of collisions on the azimuthal ion movement to be minor, since the most frequent type of collision under these conditions are expected to be ion–ion Coulomb collisions [20] and the mean azimuthal ion velocities were found to be rather similar in the experiment (compare figure 4. Further details on the simulation can be found in the appendix.

The mean azimuthal velocity is extracted from the converged simulation results for z-positions from z = 0 mm up to z = 25 mm in steps of $\Delta z$ = 0.25 mm by integrating the VDF of all particles within the defined interval [z, z+$\Delta z$]. A comparison of the simulation with the measurements is shown in figure 8. The simulation yields an increase of the azimuthal velocity in the vicinity of the target and a decrease at large distances from the target with a maximum at a distance of about 10 mm, in good agreement with the experiment. However, the simulation predicts a slightly smaller maximum velocity of only 0.95 km s⁻¹, compared to the 1.25 km s⁻¹ found in the experiment. Furthermore, the simulation shows a much less steep decrease in azimuthal velocity for z > 12 mm than the experiment.

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For the difference in maximum velocity, we propose the influence of spokes as a possible reason. Since spokes posses strong azimuthal electric fields, they should be expected to additionally affect the azimuthal velocities. However, due to the complexity of spokes, an implementation of this influence within the simulation was not possible, since the wave phenomenon propagating in azimuthal direction breaks the symmetry of the 1d simulation. Based on the good agreement between simulation and experiment, the influence of spokes appears to be smaller than the drag force between electrons and ions, caused by Coulomb collisions. However, it should be noted, that this conclusion is not necessarily valid for all discharge conditions, since spokes under the present conditions have been observed to not be very strong [41].

The difference between simulation and experiment for z > 12 mm is likely caused by collisions. In the experiment, titanium ions at these target distances experience collisions with the background gas, which will lead them to slow down. Such collisions, however, are not included in the simulation, but a slight decrease in azimuthal velocity for z > 12 mm is nevertheless reproduced. In the absence of any forces, no change of azimuthal velocity should take place in this region. However, the velocity decrease is assigned to the filtering of particles by the electric field in z-direction: only particles with a certain minimum starting velocity can reach a certain z-distance. The higher this minimum starting velocity the smaller the transit time within the region of high azimuthal force. This consequently leads to less azimuthal acceleration for the particles which are able to reach larger distances.

Despite these differences, the agreement between the simple simulation and the experiment is surprisingly good. Based on this agreement, we propose that at least a considerable part of the ion acceleration in azimuthal direction, observed in HiPIMS plasmas, is caused by the drag force from the drifting electrons on the ions via Coulomb collisions. This explanation differs from those found in the literature, which proposed different wave phenomena (the modified two stream instability or spokes) as the reason for the ion acceleration. It is likely, that these wave phenomena will also play a role in the azimuthal acceleration of ions. However, electron-ion collisions can clearly not be neglected.