HiPIMS optimization by using mixed high-power and low-power pulsing

The first goal is to derive an expression for the energy cost per ion of the film-forming species that is deposited onto the substrate. Ions of the sputtered target species are here assumed to be produced only within the dense plasma (the IR) that is maintained during the HiPIMS pulses. The total electric energy delivered to the discharge during one pulse is

$$\begin{align}\hfill {W}_{\mathrm{E},\mathrm{H}\mathrm{i}\mathrm{P}}& ={\int }_{{t}_{\mathrm{H}\mathrm{i}\mathrm{P}}}{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}{I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\enspace \mathrm{d}t\hfill \\ \hfill & ={V}_{\mathrm{H}\mathrm{i}\mathrm{P}}{\int }_{{t}_{\mathrm{H}\mathrm{i}\mathrm{P}}}{I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\enspace \mathrm{d}t\hfill \\ \hfill & ={V}_{\mathrm{H}\mathrm{i}\mathrm{P}}\left\langle {I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\right\rangle {t}_{\mathrm{H}\mathrm{i}\mathrm{P}},\hfill \end{align} \tag{ 1 }$$

where the HiPIMS power source is assumed to keep the voltage V_HiP constant throughout the pulse and the integration is taken over the HiPIMS pulse only. To simplify, we will now also assume that 〈I_HiP〉 is given by the total ion current at the target surface, neglecting the small contribution from secondary electrons [24, 31]. Note that this approach applies to an arbitrary discharge current waveform.

We also need to estimate the number of ions, of the target species, that are deposited onto the substrate from each HiPIMS pulse. These originate as atoms, sputtered by ions bombarding the target during a pulse, that are subsequently ionized. The total number of ions bombarding the target during a pulse is 〈I_HiP〉t_pulse/e, with e being the elementary charge. As a simplification, we here assume that the ions are only singly charged. In a HiPIMS pulse the effective sputter yield depends on the ion composition in the ion current onto the target. Following Hajihoseini et al [14], we define ζ to be the working gas fraction of these ions and (1 − ζ) the sputtered species fraction. In our case, with Ar working gas and a Ti target, we have an effective sputter yield Y_sput,eff per ion given by

$$\begin{equation}{Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{e}\mathrm{ff}}=\zeta {Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{A}{\mathrm{r}}^{+}\to \mathrm{T}\mathrm{i}}+\left(1-\zeta \right){Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{T}{\mathrm{i}}^{+}\to \mathrm{T}\mathrm{i}},\end{equation} \tag{ 2 }$$

where the indices are such that Ar⁺ → Ti denotes Ar⁺ as the bombarding ion species and Ti as the target material and Ti⁺ → Ti denotes self-sputtering of the target. The number of sputtered atoms per HiPIMS pulse is then

$$\begin{equation}{N}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{H}\mathrm{i}\mathrm{P}}=\frac{{Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{e}\mathrm{ff}}\langle {I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\rangle {t}_{\mathrm{p}\mathrm{u}\mathrm{l}\mathrm{s}\mathrm{e}}}{e}.\end{equation} \tag{ 3 }$$

A fraction α_t of this number becomes ionized, and of these ions a fraction β_t is back-attracted to the target. Consequently the number of target species ions per pulse that enters the diffusion region, where the substrate is placed, is N_sput,HiP α_t(1 − β_t). The number of ions that is finally deposited onto the substrate per HiPIMS pulse becomes

$$\begin{equation}{N}_{\mathrm{s}\mathrm{u}\mathrm{b},\mathrm{H}\mathrm{i}\mathrm{P}}={\xi }_{\mathrm{t}}{N}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{H}\mathrm{i}\mathrm{P}}{\alpha }_{\mathrm{t}}\left(1-{\beta }_{\mathrm{t}}\right),\end{equation} \tag{ 4 }$$

where ξ_t is the transport parameter as defined above. The energy cost per target species ion that is deposited onto the substrate is obtained by combining equations (1)–(4) to give

$$\begin{equation}{\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}=\frac{{W}_{\mathrm{E},\mathrm{H}\mathrm{i}\mathrm{P}}}{{N}_{\mathrm{s}\mathrm{u}\mathrm{b},\mathrm{H}\mathrm{i}\mathrm{P}}}=\frac{e{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}}{{Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t}\cdot \mathrm{e}\mathrm{ff}}{\xi }_{\mathrm{t}}{\alpha }_{\mathrm{t}}\left(1-{\beta }_{\mathrm{t}}\right)},\end{equation} \tag{ 5 }$$

where W_E,HiP is the electric energy delivered to the HiPIMS pulse as a whole, i.e., it also includes for example the energy used to produce the neutral flux component. All of this energy is necessary for the pulse even if we are focusing on the ions—and therefore, for simplicity, we refer to ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ as the 'energy cost' for the ions. For ion energies typical for magnetron sputtering discharges we can make the simplification that the sputter yields are approximately proportional to the square root of the ion energy [32–34]. We further assume that most of the applied potential falls over the cathode sheath, so that the ion energies are proportional to V_HiP. The sputter yields can then be written as

$$\begin{equation}{Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{A}{\mathrm{r}}^{+}\to \mathrm{T}\mathrm{i}}={K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{A}{\mathrm{r}}^{+}\to \mathrm{T}\mathrm{i}}\sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}}\end{equation} \tag{ 6 }$$

for sputtering by the working gas ions and for self-sputtering

$$\begin{equation}{Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{T}{\mathrm{i}}^{+}\to \mathrm{T}\mathrm{i}}={K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{T}{\mathrm{i}}^{+}\to \mathrm{T}\mathrm{i}}\sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}},\end{equation} \tag{ 7 }$$

where ${K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{A}{\mathrm{r}}^{+}\to \mathrm{T}\mathrm{i}}$ and ${K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{T}{\mathrm{i}}^{+}\to \mathrm{T}\mathrm{i}}$ are material-dependent coefficients specific to these two ion/target combinations. Combining equations (2), (6) and (7) gives

$$\begin{align}\hfill {Y}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{e}\mathrm{ff}}& =\zeta {K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{A}{\mathrm{r}}^{+}\to \mathrm{T}\mathrm{i}}\sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}}+\left(1-\zeta \right){K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{T}{\mathrm{i}}^{+}\to \mathrm{T}\mathrm{i}}\sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}}\hfill \\ \hfill & ={K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{e}\mathrm{ff}}\sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}},\hfill \end{align} \tag{ 8 }$$

where K_sput,eff is the effective sputter yield coefficient

$$\begin{equation}{K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{e}\mathrm{ff}}=\zeta {K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{A}{\mathrm{r}}^{+}\to \mathrm{T}\mathrm{i}}+\left(1-\zeta \right){K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{T}{\mathrm{i}}^{+}\to \mathrm{T}\mathrm{i}}.\end{equation} \tag{ 9 }$$

Combining equations (5) and (8) finally gives

$$\begin{equation}{\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}=\frac{e\sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}}}{{K}_{\mathrm{s}\mathrm{p}\mathrm{u}\mathrm{t},\mathrm{e}\mathrm{ff}}\hspace{2pt}{\xi }_{\mathrm{t}}{\alpha }_{\mathrm{t}}\left(1-{\beta }_{\mathrm{t}}\right)}.\end{equation} \tag{ 10 }$$

This is the figure of merit, which gives the energy per ion of the film-forming material deposited onto the substrate. A discharge that is arranged so that ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ is minimized will deliver ions to the substrate at the lowest possible energy cost per ion. We can note here that this expression contains only one external parameter, V_HiP, which is easily obtained in a studied discharge. The remaining four parameters α_t, β_t, ξ_t and K_sput,eff are internal to the discharge, and more difficult to assess. Tuning of α_t and β_t using accessible process parameters will be explored in the next section, while the parameters ξ_t and K_sput,eff are addressed in section 3.

The optimization task can, at this step, be formally identified as minimizing ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. In practice, however, it is not obvious how to go about this. What we have available to vary in an experiment are not the internal discharge parameters in equation (10), but a set of external process parameters. Of these, we will in this section discuss five. Two of those parameters refer to the discharge setup, B_rt and p_gas, and the remaining three parameters define a HiPIMS pulse of the type shown in figure 1: t_HiP, t_off, and $\langle {I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\left({V}_{\mathrm{H}\mathrm{i}\mathrm{P}}\right)\rangle$; the more elaborate notation $\langle {I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\left({V}_{\mathrm{H}\mathrm{i}\mathrm{P}}\right)\rangle$ is here used for 〈I_HiP〉 since we, in connection with this figure, will discuss how 〈I_HiP〉 is selected by varying V_HiP. The question is how these five parameters influence the variables in the expression on the right-hand side of equation (10).

Figure 2 shows a flow chart model, of a type earlier used in paper I, which gives a simplified overview of the network of mechanisms that are involved in the HiPIMS deposition process. The left column contains the external process parameters that can be varied, and the right column contains the figure of merit that is to be minimized. The middle column contains the two internal discharge parameters α_t and β_t. The arrows denote the mechanisms through which one parameter influences another parameter, and their color indicates the type of influence. A green (filled) arrow marks a co-correlation, i.e., an increase of the parameter at the start of a green arrow gives an increase in the parameter at the tip, and a decrease gives a decrease. A red (unfilled) arrow marks the opposite, a counter-correlation.

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We will first go through the arrows one by one, and then discuss their interplay. We start with the three arrows on the right in the figure, those which end up at the figure of merit ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. These three arrows are determined by equation (10) and are generic in the sense that they hold for any HiPIMS discharge, not only the Ti/Ar system studied here.

Influence ${V}_{\mathrm{H}\mathrm{i}\mathrm{P}}\to {\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. The pulse voltage V_HiP is not an independent process parameter but used to obtain the desired discharge pulse current. This is the reason for the notation 〈I_HiP(V_HiP)〉 in figure 2. However, a change in V_HiP also leads to a change in the sputter yield. The energy scaling of the sputter yield in equation (8) is such that it gives the proportionality ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}\propto \sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}}$ in equation (10). An increase in V_HiP thus gives an increase in ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$, and we here get a green (filled) arrow.

Influence ${\alpha }_{\mathrm{t}}\to {\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. According to equation (10) an increase in α_t gives a decrease in ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ by the same factor. Please note that no physics is involved here. This relation follows directly from the definition of the ionization probability α_t. We get a red (unfilled) arrow.

Influence ${\beta }_{\mathrm{t}}\to {\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. In equation (10) an increase in β_t gives a decrease of the factor $\left(1-{\beta }_{\mathrm{t}}\right)$ in the denominator, and therefore an increase in ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. Typical values of β_t for the studied Ti/Ar system are found to be in the range 0.6–1 [18]. Note that ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ is very sensitive to changes in β_t since it appears in the form $\left(1-{\beta }_{\mathrm{t}}\right)$ in the denominator in equation (10). We get a green (filled) arrow. Again, no physics is involved; this follows from the definition of the back-attraction probability β_t.

For the nine mechanisms represented by the remaining arrows in figure 2, which all start in the left column, the situation is different. Here the complicated and incompletely understood HiPIMS discharge physics comes into play. The colors of these arrows therefore strictly apply only to the Ti/Ar system studied here, being based on experimental data and discharge modelling. Also, they are established only within the parameter ranges studied: B_rt from 10 to 25 mTesla, p_gas from 0.5 to 2 Pa, t_HiP from 40 to 400 μs, and peak current densities at the target from 0.7 to 2.5 A cm⁻². The reader is referred to paper I for a description of the methods of analysis.

Five of the nine mechanisms have been discussed in detail in paper I and by Rudolph et al [15], i.e., those represented by the arrows 〈I_HiP〉 → α_t, B_rt → β_t, p_gas → β_t, t_HiP → α_t and t_HiP → β_t in figure 2. The analyses identifying these co- and counter-correlations are found in these papers. The remaining four mechanisms are:

Influence 〈I_HiP〉 → β_t. This is the first of two mechanisms, which were not analyzed earlier, where we have extended the study begun in paper I, as detailed in the appendix A. We find a clear trend such that an increased HiPIMS discharge current results in a lower back-attraction probability β_t. This is a very important result, since a high β_t has been identified as a major reason for the low deposition rate in HiPIMS [17, 18, 35]. Since a higher discharge current is obtained by a higher applied negative voltage to the target, this influence is somewhat unexpected: one might have expected that increasing 〈I_HiP〉 by increasing V_HiP leads to a stronger back-attraction of the positive ions. This is, however, not necessarily true, for a reason revealed by computational modeling of HiPIMS discharges by Huo et al [36]. They found that an increased discharge voltage penetrates only a little into the IR. The increase in discharge voltage instead falls almost completely over the cathode sheath, and therefore should not significantly increase the ion back-attraction from the IR. The question remains, however, why there is a reduction in β_t. Here we tentatively propose that a higher discharge currents may favor some mechanism which helps the ions to escape from the potential trap. One possibility is the acceleration in the electric field structures that are associated with spokes [37, 38]. If the spokes become stronger at higher discharge current, they might kick out ions more efficiently. In addition, Held et al [39] claim that the electric field pointing toward the target is reduced in the presence of a spoke and therefore reduces the potential barrier the ions have to overcome to escape the IR. This, however, is only a speculation and the anti-correlation between the discharge current and β_t is to be regarded as a model result based on experimental input. It is in figure 2 represented by a red (unfilled) arrow.

Influence p_gas → α_t. This is the second mechanisms for which we have extended the data analysis from paper I. The modelling of the studied discharges reveals a significant effect from the pressure on the electron energy distribution function (EEDF), which is also supported by measurements of the effective electron temperature in the same discharges [40]. When the pressure decreases from p_gas = 2 Pa to p_gas = 0.5 Pa, the modeled electron temperature of the thermal electron population increases from T_e ≈ 4 eV to the range 5.5–6 eV [41]. This has a dramatic effect on the electron impact ionization rate coefficient k_iz, which, for the reaction Ti + e → Ti⁺ + 2e, increases by a factor of 2. In addition, a lower pressure gives a larger fraction of doubly ionized Ti²⁺ ions which, in contrast to Ti⁺ ions, emit secondary electrons from a Ti target upon bombardment (data given in [41]). The secondary electrons are accelerated across the cathode sheath and build up a hot electron population in the plasma. Thus, a lower working gas pressure boosts the hot electron population. These two effects of the lower pressure (higher electron temperature and increased secondary electron emission) both act in the same direction, towards a more energetic EEDF and a higher ionization rate. The arrow from p_gas to α_t is drawn by a red (unfilled) arrow in figure 2. Furthermore, a low working gas pressure has the additional advantage of decreasing the average number of collisions that the energetic plasma species, both ions and neutrals, undergo on their way to the substrate [42, 43] and subsequently increases the energy transferred to the growing film [44, 45]. It is for example well established that lower working gas pressure in HiPIMS leads to denser films and lower surface roughness [46].

The two remaining mechanisms to be assessed are the influences t_off → α_t and t_off → β_t. For these we have not yet any experimental data but can make the following considerations on what is a likely trend. The key is the fate of the 'reservoir' of ions, those that are left in the magnetic trap at the end of the HiPIMS pulse [15]. During the off-time no back-attracting electric field exists. The ions in the reservoir can therefore escape easier towards the substrate, corresponding to a lower β_t. Since both sputtering and electron impact ionization cease abruptly at the end of the pulse, neither new atoms nor new ions are created, and therefore α_t is not changed. When the dcMS-like voltage V_dcMS is applied at the end of the off-time (figure 1), back-attraction sets in again for the ions remaining in the reservoir. A shorter t_off therefore corresponds to back-attraction of a larger fraction of the ions that were produced during the HiPIMS pulse, i.e., a higher effective β_t. In figure 2 we get a red (unfilled) arrow t_off → β_t and no arrow t_off → α_t.

Having determined the type of influence (the colors of the arrows) of the mechanisms in figure 2, we now turn to their interplay. The color coding of the arrows makes it easy to assess how the change of a process parameter influences the figure of merit ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. Let us start with the three parameters B_rt, p_gas and t_HiP. Two arrows of the same color, along a route, means that an increase in the initial process parameter leads to an increase in ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$, and a decrease to a decrease. This is the case for both routes from p_gas to ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. A lower working gas pressure therefore leads to a lower cost per ion onto the substrate, which is desired. For the two routes from t_HiP to ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ the situation is more complicated: the route through α_t has arrows of different colors (a counter-correlation), and the route through β_t the same color (a co-correlation). To quantify which route dominates, we have here extended the model study by Rudolph et al [15] on the effect of the pulse length. The result is shown in figure 3. The product α_t(1 − β_t) increases for shorter pulses. According to equation (10), this gives a lower cost of ionization ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ for shorter pulses. Finally, from B_rt to ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ there is only one route, with two arrows of the same color, indicating a co-correlation. A lower magnetic field strength reduces ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. For these three parameters the optimization rules, not surprisingly, become the same as those proposed in paper I: weak B_rt, low p_gas and short t_HiP. The remaining process parameter t_off was not discussed in paper I. Here we have two arrows of opposite color, a counter-correlation. A longer t_off therefore reduces ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. We note, however, that the process of emptying the ion reservoir should not take much more than the time τ_empty ≈ l_c,IR/v_i,slow, where l_c,IR is a characteristic extent of the IR and v_i,slow is the speed of the slowest ions. Off-times longer than τ_empty would give little extra benefits. For Ti, this time lies somewhere between 10 μs and 250 μs, where the lower limit is derived assuming an average kinetic energy of the ionized sputtered species of 15–20 eV [42, 47] and l_c,IR = 0.05 m. The upper limit estimate is based on the thermal velocity of the ionized sputtered neutrals (keeping in mind that only a minority of the ions can be expected to be completely thermalized) and the same characteristic length l_c,IR. The speed v_z in the direction away from the target corresponds to one third of the average energy 3k_B T_gas/2 per particle. Using ${m}_{\mathrm{i}}{v}_{z}^{2}/2={k}_{\mathrm{B}}{T}_{\mathrm{g}\mathrm{a}\mathrm{s}}/2$ gives a typical speed of ${v}_{z}=\sqrt{{k}_{\mathrm{B}}{T}_{\mathrm{g}\mathrm{a}\mathrm{s}}/{m}_{\mathrm{i}}}\approx$ 227 m s⁻¹, where k_B is Bolzmann's constant, and the numerical value is evaluated for 300 K and for the titanium mass.

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Let us now consider the interplay between the five arrows in figure 2 that are related to the average discharge current during the HiPIMS pulses 〈I_HiP〉. Two routes from 〈I_HiP〉 to ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ go through α_t and β_t. Both routes are represented by arrows of different color in figure 2, showing an anti-correlation. Higher 〈I_HiP〉 should, if these were the only mechanisms, give a desired lower ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$. However, in order to increase 〈I_HiP〉, one has to increase the pulse voltage. The discharge current in a HiPIMS pulse is in practice regulated by the applied pulse voltage V_HiP, but the relation is far from simple. As pointed out by Anders et al [48], a voltage–current–time representation is generally necessary to describe the discharge properties. The discharge current time profile is often triangular for short pulses, while for longer pulses it saturates at a plateau, sometimes at a level below a transient discharge current maximum [3]. The general trend, however, is that a higher peak discharge current is obtained at a higher pulse voltage. This is the only observation we need here for a qualitative discussion. Since the higher voltages, that are needed for a higher discharge current, increases ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ (the direct green arrow from V_HiP to ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ in figure 2), there are two conflicting influences that connect 〈I_HiP〉 to ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$: a route of co-correlation through V_HiP, and two routes of counter-correlation through α_t and β_t. In order to assess in which direction the net effect goes, we use experimental data from Lundin et al [40] in which the peak current density, the working gas pressure, and the pulse length were varied. The values of α_t and β_t have been determined by modelling as described in paper I. An extract from a larger data set is shown in table 1. In order not to overburden this table, only one pulse length (t_HiP = 100 μs) out of two, only one working pressure (p_gas = 0.5 Pa) out of two, and only two peak discharge current densities (J_HiP,peak = 0.7 A cm⁻² and 2.5 A cm⁻²) out of three are shown. We are here only interested in the trends, and these are the same for the other combinations. We see that an increase in V_HiP from 450 V to 500 V leads to a peak current density increase from 0.7 A cm⁻² to 2.5 A cm⁻². The direct route ${V}_{\mathrm{H}\mathrm{i}\mathrm{P}}\to {\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ increases the latter with a factor $\sqrt{500/450}$, but this is only 5%. The two opposing routes, $\langle {I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\rangle \to {\alpha }_{\mathrm{t}}\to {\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ and $\langle {I}_{\mathrm{H}\mathrm{i}\mathrm{P}}\rangle \to {\beta }_{\mathrm{t}}\to {\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$, both give opposing (decreasing) factors. The latter dominate as shown by the combined factor given in the bottom row of table 1, which is taken from equation (10). Increasing the peak current density from 0.7 A cm⁻² to 2.5 A cm⁻² gives a lower energy cost ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$ by a factor 129/203, i.e. a reduction by 36%. We conclude that the negative effect of a less energy-efficient sputter yield for a higher V_HiP is more than counteracted by the positive effects of increasing α_t and decreasing β_t at a higher 〈I_HiP〉. The net effect is that the highest possible pulse discharge currents should be the goal, which was not expected a priori.

Table 1. Numerical values used in the discussion of the effect of varying the HiPIMS pulse discharge current amplitude. The data is taken from the experiments by Lundin et al [40] and the analysis by Butler et al [17, 41].

pgas (Pa)	0.5
tHiP (µs)	100
JHiP,peak (A cm−2)	0.7	2.5
VHiP (V)	450	500
αt	0.87 ± 0.01	0.96 ± 0.01
βt	0.88	0.82
αt βt	0.766	0.787
$\frac{\sqrt{{V}_{\mathrm{H}\mathrm{i}\mathrm{P}}}}{{\alpha }_{\mathrm{t}}\left(1-{\beta }_{\mathrm{t}}\right)}$	203	129

The sixth row in table 1 gives the product α_t β_t. It was pointed out by Čapek et al [9] that this combined parameter quantifies the return effect, in the sense that it gives a direct measure of the loss in deposition rate that can be attributed to ion back-attraction: a fraction α_t of the sputtered atoms becomes ionized, and a fraction β_t of these ions return to the target. There is, however, a drawback with using this product alone. The data in table 1 reveals that, in our case, an almost constant value of the product α_t β_t (only 3% difference) hides two separate, and desired, trends with increasing discharge current: a higher α_t and a lower β_t. This shows the importance of considering α_t and β_t separately, and not only the product. It can be noted here that Bradley et al [10] found similarly small changes in α_t β_t with discharge current, in the range 3%–13%, when the discharge current was varied by a factor of about two.

Table 2 summarizes the results so far concerning how to minimize ${\mathbf{\varepsilon }}_{\mathrm{t}\mathrm{i},\mathrm{H}\mathrm{i}\mathrm{P}}$, both the goals for the process parameters and brief summaries of the physical mechanisms involved. Please keep in mind that these results are based on experimental data for the Ti/Ar system only, and for limited parameter ranges: B_rt down to 10 mTesla, p_gas down to 0.5 Pa, t_HiP down to 40 μs, and J_HiP,peak up to 2.5 A cm⁻². The observed trends at these limits are favorable, however, and it should be worthwhile to extend experiments beyond these values.