Particle-in-cell/Monte Carlo collisions treatment of an Ar/O2 magnetron discharge used for the reactive sputter deposition of TiOx films

The physical processes in an Ar/O2 magnetron discharge used for the reactive sputter deposition of TiOx thin films were simulated with a 2d3v particle-in-cell/Monte Carlo collisions (PIC/MCC) model. The plasma species taken into account are electrons, Ar+ ions, fast Arf atoms, metastable Arm* atoms, Ti+ ions, Ti atoms, O+ ions, O2+ ions, O− ions and O atoms. This model accounts for plasma–target interactions, such as secondary electron emission and target sputtering, and the effects of target poisoning. Furthermore, the deposition process is described by an analytical surface model. The influence of the O2/Ar gas ratio on the plasma potential and on the species densities and fluxes is investigated. Among others, it is shown that a higher O2 pressure causes the region of positive plasma potential and the O− density to be more spread, and the latter to decrease. On the other hand, the deposition rates of Ti and O are not much affected by the O2/Ar proportion. Indeed, the predicted stoichiometry of the deposited TiOx film approaches x=2 for nearly all the investigated O2/Ar proportions.


Introduction
Magnetron discharges have widespread applications in the field of 'sputter deposition' of thin metallic or compound films. In this process, particles with a sufficient amount of energy bombard a target, which initiates a collision cascade in the target, leading to a possible release of a surface atom. This target process is called 'sputtering'. The sputtered atoms pass through the reactor chamber and can be deposited on a substrate, placed in front of the target, forming a thin metallic coating. In our case, the energetic particles originate from argon background gas, which is electrically transformed into a plasma: by coupling an electrical circuit to the target, and putting the other walls at ground potential, the target acts as a cathode and the other walls as anode. Hence, an electric field is generated, which partially breaks down the gas into ions and electrons, i.e. a plasma is created.
The plasma is magnetically enhanced: by placing magnets behind the cathode target, a magnetic field is created. Charged particles from the plasma circulate around the magnetic field lines with their Larmor radius: The high ion mass causes the ion Larmor radius to be very large, leading to an almost straight movement of the ions. However, electrons have a much lower mass, which causes them to be trapped in the magnetic field. As a consequence, the electrons have longer mean free paths, and therefore a higher probability for ionizing the background gas. When these ions, accelerated by the electric field, hit the cathode target, target material can be sputtered. The enhanced ionization results in a large amount of sputtering, and allows the magnetron discharge to operate at lower pressures (typically in the range of 0.1-1 Pa [1]) in comparison with non-magnetized sputtering discharges. Due to the lower pressure, the sputtered atoms will be less scattered on their way to the substrate. This results in a more effective deposition. Most magnetized electrons reside where the radial magnetic field is at a maximum. As a consequence, the ionizations occur in the same area. In the present work, a planar circular magnetron in a cylindrical chamber is investigated, meaning that the magnetron consists of a planar circular cathode on which a planar circular target is clamped. The magnetron is balanced, meaning that the majority of the magnetic flux lines that originate at the cathode, also return to it without crossing the anode. In a planar balanced magnetron, the maximum plasma density is therefore found in a narrow area in front of the cathode (denoted as a dotted area in figure 1). Therefore, the planar target is locally sputtered, leaving behind an erosion cavity, the so-called 'race-track'.
When a reactive gas, like oxygen, is added to the Ar background gas, the O 2 gas molecules themselves as well as atoms originating from the O 2 gas (by plasma reactions, or by sputtering a poisoned TiO x target) can react with the sputtered metal atoms on the substrate to form a metal oxide layer [2]- [4], in a process called 'reactive sputter deposition'. This process is schematically presented in figure 1. 4 and numerical models is desirable to provide a complete chemical and physical picture of magnetron discharges, needed to optimize the applications.
Different kinds of models exist to simulate gas discharges. Mostly, these models are subdivided into analytical, continuum and particle models, as well as hybrid models which comprise combinations of these aforementioned models.
'Analytical models' are based on simple analytical (mostly (semi-)empirical) formulae to describe the behavior of macroscopic plasma characteristics, such as voltage, current, pressure and magnetic field. These models are very fast, but they are limited in accuracy (because approximations are used) and generality (for example, they are only valid for discharge conditions in specific ranges). For magnetron discharges, analytical models exist to predict quickly some general relations between macroscopic plasma characteristics [9]- [15]. The reactive sputter deposition process is modeled in [5,16,17], by means of balance equations of the particle fluxes toward these surfaces.
'Continuum or fluid models' are based on the continuity equations and calculate the electric field in a fast and self-consistent way. It is hence a very powerful modeling approach, if the 'local field approximation' is fulfilled, i.e. the charged particles' energies must be in equilibrium with the electric (and magnetic) field. In other words, the energy loss by the collisions must balance the energy gain by the fields, and the plasma must behave as a continuum or a fluid. Fluid models are widely used to simulate gas discharges; however, modeling magnetron discharges with a fluid model is not so common, since they cannot be simply considered as a fluid. Indeed, in a low pressure discharge, such as a magnetron discharge, the loss of energy caused by collisions is much smaller than the energy gain due to the electric field, especially for electrons. Furthermore, the complexity of the magnetic field makes a continuum model for magnetron discharges very inefficient [18]. Nevertheless, in some papers [19,20] magnetron discharges are simulated by fluid models.
'Hybrid models' denote mostly the combination of Monte Carlo (MC) or particle-incell model/Monte Carlo collisions (PIC/MCC) descriptions for the fast electrons, and fluid approaches for all the other species (slow electrons, ions and neutrals). However, the same difficulties arise as for fluid models.
'Particle models' treat every species individually, and hence these models do not have to obey the condition of continuum models. An example is an MC model [21], which treats the collisions probabilistically, and calculates the particles' movements, starting from an initially defined electric field distribution. This modeling technique is rather fast, because the electric field is not calculated self-consistently from the external electrical circuit and the spatial distribution of the charged particles. However, to make the MC model self-consistent, this model can be coupled to a so-called PIC model [22]- [24], which calculates the electric field in a self-consistent manner. The coupled model is named PIC/MCC model [23,24], and is able to produce a wealth of data, such as cathode voltage and current, plasma species densities, fluxes and energies, potential distribution and information on separate collisions. PIC/MCC models calculate the entire discharge behavior very accurately; however, this goes in expense of a longer calculation time, which is partially accounted for by representing real particles by a limited number of superparticles (SPs), and by weighting the SPs on a grid (hence the name 'PIC'). However, permanently evolving computational devices (fast computers) and methods (code parallelizing) make these methods nowadays very attractive.
Magnetron sputter deposition is investigated by MCC models [25], and more precisely, the reactive sputter deposition was studied in [26]. PIC/MCC models for sputter deposition in 5 a pure Ar discharge are extensively developed [23,24], [27]- [32]. To the authors' knowledge, one PIC/MCC model exists for an Ar/O 2 gas mixture [33]. However, this PIC/MCC model does not take into account plasma-surface interactions, such as target sputtering, target poisoning, and atom sticking, i.e. the sputter deposition process itself is not described. Moreover, these plasma-surface interactions influence all of the calculated discharge characteristics [6,7]. Secondly, the external circuit is not included in the model of [33]. Nevertheless, the external circuit occurs to be inevitable in a PIC/MCC code for an accurate and correct description of magnetron discharges [34].
In our opinion, the PIC/MCC modeling approach produces the most accurate data, making it a very powerful tool to simulate magnetron discharges. However, a complete PIC/MCC model does not yet exist to describe the reactive sputter deposition in an Ar/O 2 gas mixture in a magnetron discharge. Therefore, to study the reactive magnetron sputter deposition process of TiO x layers, a PIC/MCC model including plasma-surface interactions is developed and presented in this paper.
In the present work, the effect of the O 2 /Ar ratio on I-V characteristics, on the plasma potential distribution and on different plasma species densities is calculated. Moreover, sputtered Ti and O fluxes from the target are calculated for different O 2 /Ar proportions, as well as Ti, O and O 2 fluxes to the substrate. The coupled analytical surface model enables to describe the deposition process of the TiO x film, in terms of the deposited TiO x fraction, the Ti and O deposition rates and the film stoichiometry x.

Description of the model
In the PIC/MCC model, the particle movement is simulated with the PIC method, and the collisions are treated with the MCC module. Plasma-surface interactions, such as sputtering, electron emission, species reflection and the effect of a poisoned target, are accounted for in our model. A brief overview, with definitions of certain characteristics and coefficients, is given here. Overviews of the PIC/MCC model can be found in [18], [22]- [24], [32,34]. A detailed description of a similar model developed for an Ar/N 2 discharge can be found in [35].
The species present in the discharge (electrons, Ar + ions, fast Ar f atoms, metastable Ar * m atoms, Ti + ions, Ti atoms, O + ions, O + 2 ions, O − ions and O atoms) are represented by SPs with a corresponding weight factor, describing the amount of real particles per SP. In the PIC algorithm, the properties of the different particles, such as the densities, are calculated on a grid. Weighting the particles' positions to a grid gives the charged particle density on the grid. The potential distribution V is calculated from the densities of the negative species (electrons and negative ions) n − and the densities of the positive ions n + : This equation is solved with the superposition principle, presenting the potential V as the sum of the potential only due to the space charge, V P , and the potential only due to the cathode voltage, U 0 , i.e. V = V P + U 0 V L . U 0 is calculated from the coupling of the external circuit to the plasma, and V L is the dimensionless potential caused by an applied voltage with magnitude 1 V. A detailed description can be found in [34]. The new electric field distribution E is calculated from V : 6 After weighting the electric field on the grid back to the particles' positions, the motion of the charged SPs in the electric and magnetic fields are calculated with Newton's equations: Note that the electric field is both externally applied as well as created by the charged SPs themselves, whereas the magnetic field is only externally applied. The neutrals are not influenced by the present electric and magnetic fields. The SPs in the discharge can also interact with each other, described in the MCC algorithm. The different plasma reactions can be subdivided into elastic and inelastic collisions. Depending on the type of collision, the total kinetic energy is conserved (elastic collisions) or not (inelastic collisions). In the latter case, the energy is partly consumed as internal chemical energy for one of the collision partners (such as excitation, ionization, dissociation and dissociative ionization), or created and supplied to the discharge as light or heat (such as deexcitation and recombination). The probability of occurrence of the various collisions is dependent on the speed of the incident particle (v), the density of the target species (n tar ) and a collision probability, which is based on the collision cross section (σ ), or on the reaction rate constant (k). The collision probability for collision j is presented as Some cross sections of ion-neutral collisions are described with the Langevin-Hasse model [36,37], assigned 'LH' in table 1, since no other data were available in the literature. This approach calculates the reaction cross section, based on the dimensionless impact parameter for which the deflection angle is negligibly small [38] (assumed 3 for all species [37]), and on the masses, velocities and polarizability of the colliding species (11.08 a 3 0 for Ar [39], 10.60 a 3 0 for O 2 [39] and 5.4 a 3 0 for O [39], where a 0 is the Bohr radius). After the collision, the SPs are scattered from their original part, and their new velocities (direction and magnitude) are based on the center of mass theory [40], with a characteristic scattering angle, which is dependent on the type of collision. This scattering angle is mostly isotropic in all directions, except for electrons, where the scattering angle depends on the energy of the electron. Further details about their treatment for a magnetron discharge in argon are given in [23,32]. The collisions included in the model are summarized in table 1. Note that certain species that are present in the discharge are not followed in the model, since they are not very relevant, such as O 3 (which is created by reactions (49) and (50)).
Apart from interactions in the plasma, SPs can also interact with the reactor walls, after which reflection (and/or neutralization), adsorption or the creation of new SPs can occur. The latter can be the creation of secondary electrons, whose yield is characterized by the secondary electron emission coefficient (SEEC), or sputtered atoms, determined by the sputter yield formula of Matsunami et al [41]. Reflection of electrons is determined by the reflection coefficient, whereas heavy particles reflection is characterized by the sticking coefficient (SC). If a particle's SC is zero, the particle is reflected when hitting a wall. For Ar atoms, an SC of zero is assumed, since Ar is an inert gas. All included ions are assumed to neutralize after reflection at the walls, which implies that they also have a zero SC, since they are reflected, Table 1. List of the collisions considered in the model. The references where the cross sections (σ (E)) or rate constants (k) were adopted from are presented in the last column. The rate constants are in m 3 s −1 for two-body collisions and m 6 s −1 for three body collisions (indicated by * ). '(a)' refers to assumed rate constants based on similar reactions, due to lack of data in the literature. 'LH' refers to the Langevin-Hasse treatment for the cross section (see section 2).
Recombination k = 2 × 10 −13 (300/T gas ) 0.5 [63] albeit, as neutral particles. Moreover, the metastable Ar * m atoms are reflected as ground state Ar atoms. In other words, the reactor walls act as a loss mechanism for Ar + ions, Ar * m metastable atoms, Ti + ions, O + ions, O + 2 ions and O − ions. In contrast to the above-mentioned species, Ti and O atoms, and O 2 molecules do have a certain probability to interact with a wall, i.e. the sticking probability, or SC, is not zero. In the low pressure case that we investigate, the SC does not have a large impact on the corresponding particle densities. Therefore, on the walls, excluding the substrate, a constant SC is assumed, i.e. 0.5 for Ti [42], 1 for O [5] and 0 for O 2 . However, to describe the deposition process of the TiO x film on the substrate, and to predict its stoichiometry, some effective sticking coefficients, SC eff , of Ti, O and O 2 need to be calculated. These SC eff values are not constant, but are influenced by the incorporation or coverage of the different implanted or adsorbed species on the surface, respectively [5]. Therefore, an analytical surface model, based on the models described in [16,17,43] Note that we use the static description of the analytical surface model. This means that θ t,r cs is not calculated with its value from the previous time step, i.e. dθ t,r cs /dt = 0. The time dependence arises from the PIC fluxes, F t,r , which are calculated as a function of time.
Since the desired stoichiometry x of the TiO x film is two, the surface model needs to account for this. Therefore, the compound fraction balance equation, for the deposition of a TiO x film with stoichiometry x = 2, is With this equation, at every PIC/MCC time step (t), the compound (c) fraction on every radial position (r ) on the substrate (s), i.e. θ t,r cs , is derived from the fluxes (F) of the incoming O 2 , O and Ti species (calculated with the PIC/MCC model), with corresponding constant SC, which are chosen as 0.3 for O 2 [5], 1 for O [5] and 0.5 for Ti [42]: Subsequently, the SC values of O and O 2 are adapted at that time step by the compound fraction, θ t,r cs , so that we obtain the so-called effective sticking coefficients, SC eff , for this time step, on every position r on the substrate: . Indeed, the sticking of O and O 2 on TiO x compound will be lower than on a pure metallic surface. Since the sticking of Ti is independent of the compound fraction (i.e. Ti will stick in a similar way on a metallic Ti surface and on a TiO x surface), a constant SC Ti can be used, i.e. SC t,r Ti,eff = SC Ti . Note that the actual stoichiometry of the deposited TiO x film at steady state, x t,r , is found by multiplying the desired stoichiometry, x = 2, with the compound fraction, θ t,r cs . When an SP is created or reflected at a wall, the direction of its velocity is chosen as such to obey the cosine law [44]. As a consequence, the azimuthal angle, φ = 2π.RN[0, 1], whereas the polar angle, θ = sin −1 ( √ RN[0, 1]). RN[0, 1] is a random number uniformly distributed in the interval [0, 1]. For emission of an electron from the cathode surface, the components of the velocity vector v, v r and v θ are correspondent to the azimuthal angle, φ, whereas v z is derived from the polar angle, θ. In cylindrical coordinates, this becomes: When this created or reflected SP is an electron, its energy is assumed as 4 eV. When the SP is another species, its energy depends on the thermal accommodation coefficient, α: where E i is the mean energy of the incoming particles, E r is the mean energy of the reflected particles and E w is the mean energy of the reflected particles in thermal equilibrium with the wall (2kT w ). In our simulation, α is assumed as 0.5 [45], and a random number between 0 and 1 is generated. When the random number is lower than α, an inelastic reflection occurs, and if the random number is higher than α, an elastic reflection occurs. After an inelastic reflection, the particle is thermalized and removed from the calculation (unless in the case of a reflected Ti or O atom, which is described with diffusion equations, see below). On the other hand, if the particle is elastically reflected, its energy is conserved. The direction of the velocity is chosen as such to obey the cosine law [44], and its components (v r , v θ , v z ) are as in equation (10). Also, the effects of poisoning on the plasma-target interactions, i.e. on the SEEC and on the sputter yield, is included in the model. In the present work, we used O 2 gas flows for which the target is completely in the poisoned mode, in order to avoid (i) the simulation of the hysteresis and (ii) having to deal with a partially reacted Ti target. However, due to the target surface modification the SEEC is altered, which results in a target voltage change [46]. In the model, the poisoning of the target is therefore described by changing the SEEC value, depending on the O 2 gas pressure, and in correspondence to the range reported in [46]. As far as sputtering is concerned, both Ti and O atoms can be sputtered from a poisoned target, but with a lower sputter yield than for a metallic Ti target. Therefore, in the model, the sputter yield, calculated with the formula of Matsunami et al [41], is lowered by a factor, in accordance to the values reported in [47]: the sputter yield of Ti from a fully poisoned TiO x target is lowered by a factor of 8.8 compared to the sputter yield of Ti from a metallic target, and the sputter yield of O from a poisoned target is two times higher, compared to the sputter yield of Ti from a fully poisoned target [47].
When treating the slow (thermalized) particles with the PIC/MCC method, the computation time would be tremendously high to reach a steady state. Therefore, alternative methods are used, according to the species' physical behavior. The neutral slow particles can be subdivided into motionless and diffusing species. In the plasma, the motionless background gas atoms or molecules (Ar and O 2 ) are initially slow, but can be heated [48]. The partial pressure of the gases, p Ar and p O 2 is kept constant in each simulation. Thermalization of fast atoms by collisions with background gas atoms or molecules can cause the background gases to heat up, i.e. the temperatures, T Ar and T O 2 will increase. The densities of the gases, n Ar and n O 2 , which change due to gas heating, can be calculated at every position and at every time step, with the ideal gas law: Here, k B is the Boltzmann constant. The flux of O 2 is needed to picture the deposition of the TiO x film. The flux is calculated as the amount of collisions with a surface per second: The velocity, v, is the average velocity of the gas molecules, i.e. thermal velocity v = 2k B T m . The diffusing atoms can be created as slow particles, or can be slowed down by collisions (Ti and O). Their density n slow (r, z) is calculated with the diffusion equation: where D is the diffusion coefficient of Ti or O atoms in Ar gas, and r prod and r loss are the production and loss rates. The production of a slow Ti or O atom (r prod ) can be caused by thermalization of a fast Ti or O atom, or direct creation of a slow Ti or O atom. The loss of a where T is the gas temperature (K), M 1 and M 2 are the masses of Ti (or O) and Ar (g/mole), p is the pressure (atm) and d 12 is the collision diameter (10 −10 m), given by (d 1 +d 2 ) 2 . The collision diameter of Ti is 2.684 × 10 −10 m [50], 3.050 × 10 −10 m for O [51] and 3.542 × 10 −10 m for Ar [51].

General conditions
The simulations are performed for a planar circular magnetron with a 25 mm radius Ti target, installed in a cylindrical chamber. The magnetron discharge operates at room temperature (300 K). The axisymmetric magnetic field has a maximum radial strength of 1040 Gauss, and is schematically presented in figure 2. Due to the cylindrical symmetry of the reactor, the simulations can be carried out in 2d (r,z): the simulation area is presented in figure 2.
An electrical circuit, consisting of an external resistance (R ext = 1500 ) and voltage source (V ext = −600 V ), is connected to the cathode, whereas the other walls are grounded, to generate an electric field. The coupling of the external circuit to the PIC/MCC code is explained in [34].

Potential distribution, and charged species density profiles
Light-weighted electrons have the highest mobility from all the species in the discharge. Therefore, they are rapidly lost to the reactor walls, while the (mainly positive) ions remain in the discharge. As a consequence, the walls charge negatively, leaving behind a positively charged 'sheath' area. This transition area between the bulk plasma and the wall is characterized by a decreasing potential from the bulk to the wall (by equation (2)). This causes positive ions to reside in the sheath and negative ions to reside in the positive potential part of the plasma, i.e. the bulk (see below). When a negative potential is applied to the cathode, the sheath in front of the cathode is the most distinct. Indeed, the whole potential difference is passed through in this small area. As a result, the positive ions which enter the cathode sheath are accelerated to the cathode target. When an external magnetic field is present, electrons are confined to the magnetic field lines, leading to a higher electron density at the maximum radial magnetic field, as presented in figure 3 (for the case of 0.12 Pa O 2 ). The electron density is only weakly dependent on the O 2 /Ar proportion.
The electrons ionize neutral Ar, O 2 , O and Ti species, causing the positive ion density to be highest in the same area. All have similar profiles as the electron density so they are not shown here. The Ar + density has approximately the same magnitude as the electron density, the O + and O + 2 densities are two orders of magnitude lower (the latter is shown in figure 4), and the Ti + density is three orders of magnitude lower. Both the Ar + and the Ti + density do not change much with O 2 /Ar ratio. However, the O + and O + 2 densities rise with O 2 gas pressure, as is clear from figure 4. The total positive ion density profile causes the sheath to be thinnest where the density is at maximum, i.e. at a radial position of 13.5 mm, as a consequence of decreasing Debye length at increasing ion density.
The charged particles' distribution generates an electric potential distribution, and together with the externally applied potential, a typical plasma potential distribution is created, as shown in figure 5. A steep ascend from cathode to bulk over the sheath area is observed, becoming flat and positive in the bulk, and decreasing to zero toward the grounded anode walls. Due to the external magnetic field, the plasma potential well is steeper and the sheath is thinner at the area where the radial magnetic field is at maximum, as is clear from figure 5.
When no O 2 gas is present, the negative space charge is only created by electrons, which are confined to the magnetic field in an area close to the cathode. As a result, the positive area of the electric potential is only found in a narrow area close to the grounded reactor walls, as is clear from figure 5 (top figure).
When O 2 is added to the Ar gas, negative O − ions are created by the electron impact reactions (11) and (12) from table 1. Because the electrons are confined to the magnetic field in an area close to the cathode, the O − ions are also created in this area. However, taking into account that heavy O − ions are not trapped in the magnetic field in contrast to electrons, the plasma potential distribution (figure 5) forces the negative O − ions to move towards the positive potential area. Figure 6 presents the calculated O − density, which has a maximum at the positive potential area. This negative ion density causes the positive potential area to enlarge, see figure 5 (compare 'without O 2 ' with '0.02 Pa O 2 '). Figure 5 also illustrates that at increasing O 2 density, the positive potential area moves toward the symmetry axis of the magnetron. This is also true for the O − density (see figure 6), which also becomes smaller and more spread out. The decreasing trend of the O − density is a bit unexpected, but it is a consequence of the O + and O + 2 densities, which rise significantly when the O 2 pressure is increased, as was clear from figure 4 above. Consequently, the recombination reactions (53), (54) and (55) from table 1 occur much more often, which are an important loss mechanism for O − ions. Therefore, the O − density decreases upon O 2 addition (see figure 6). When the density drops, the Debye length increases leading to a less pronounced O − shielding of the anode potential. This results in a spreading of the density profile and a spreading of the positive potential area toward the center of the discharge.

Calculated sputtered Ti and O fluxes
The plasma species can bombard the target surface and sputter Ti and O atoms. Positive Ar + , Ti + , O + and O + 2 ions, and fast Ar f atoms contribute to the sputtering of the poisoned TiO x target. The sputtered Ti and O fluxes at different values of the O 2 partial pressure are presented in figure 7. The sputtered Ti flux in the case of a pure Ar discharge is not presented here. It is around 10 times higher than the obtained valued in the O 2 /Ar mixture as a consequence of the lower sputter yield when the target is poisoned. As a consequence of localized bombardment of plasma species to the target, the sputtered Ti and O fluxes exhibit a peak profile. This localized sputtering causes the typical eroded race-track area in the target.
The sputtered Ti and O fluxes are approximately equal, although the sputter yield of O is two times higher than of Ti (see section 2). This is caused by the fact that the threshold energy for sputtering Ti by Ar + ions is lower (the main sputter source in the discharge) than sputtering O by Ar + ions (the threshold energy for sputtering Ti with Ar + ions is 12.55 eV and 18.85 eV for sputtering O). Also, there is no dependence of the amount of sputtering on the O 2 partial pressure, since most sputtering is caused by Ar + ions, which flux is independent of the O 2 partial pressure.    The Ti flux to the anode has a peak profile caused by the sputtered Ti peak profile (see figure 7). This will cause the deposited Ti in the film to be non-uniform. However, the peak is broadened as a consequence of diffusion of the sputtered Ti atoms through the plasma. This means that when the substrate is placed further from the target, the film thickness will become more uniform. The Ti flux to the anode remains virtually constant for increasing O 2 amount, in correspondence to the sputtered Ti flux (see section 3.3 and figure 7 above).
Although the sputtered Ti and O fluxes are equal, the O flux to the anode is almost everywhere higher than the Ti flux. This is caused by the additional production processes of O atoms besides sputtering, i.e. O atoms are also created by plasma reactions. Moreover, the O flux has lost its radial peak profile, caused by O sputtering (see figure 7), because more O atoms are created by plasma reactions than by sputtering. The O flux increases upon O 2 addition, which is a consequence of higher O production at higher O 2 amount.
At the gas pressures under consideration, gas heating is not so important [48]. As a result the Ar and O 2 gas densities remain constant and uniform (see also equation (12)). Consequently, the O 2 flux bombarding the substrate has a uniform spatial profile. Logically, the O 2 flux increases for higher O 2 partial pressure (see also equation (13)).

Calculated deposition of a TiO x film on the substrate
Based on these calculated Ti, O and O 2 fluxes (figure 8) and the corresponding SC values, the radial deposited fraction of TiO x (i.e. the 'compound fraction') on the substrate is calculated with equation (8), and it is presented in figure 9. It is clear that the compound fraction is close to unity for all O 2 partial pressures investigated. The lower compound coverage at a radial position of 13.5 mm is caused by the Ti flux, because both quantities are inversely proportional by equation (8). At increasing O 2 amount, the compound fraction on the substrate increases as a  Figure 11 shows that the total deposition rate of O is two times higher than the deposition rate of Ti, which is required to deposit a TiO 2 film. The calculated stoichiometries for three of the investigated O 2 partial pressures are also presented (figure 11: gray curves, right axis). At the lowest O 2 amount (0.02 Pa), the stoichiometry of two is almost achieved (with a minimum value of 1.93). At increasing O 2 amount, the deposition rates of both Ti and O are hardly affected, but nevertheless, the small effect results in a stoichiometry becoming equal to two. The latter could also be predicted by the compound coverage, as was presented in figure 9: the real stoichiometry of the film is found by multiplying the desired stoichiometry x = 2 by the compound coverage (note that the desired stoichiometry x = 2 was imposed by equation (7)). This means that an O 2 /Ar proportion of 1/10 is already enough to deposit a stoichiometric film (with a minimum value of 1.98), with a satisfactory deposition rate.
The peak profiles in figure 11 for the total deposition rates of Ti and O illustrate that the deposited film will have a non-uniform thickness, under the investigated conditions. This is caused by the non-uniform Ti flux (figure 8) leading to radially dependent SC eff values (figure 10). It is expected that a more uniform film can be deposited by placing the substrate further from the target so that the fluxes to the substrate are smoothed out. It was also found that O 2 contributes at least 50 times more than O to the total deposition of oxygen in the TiO x film. This is caused by the much higher O 2 flux compared to the O flux (see figure 8 above), although the SC eff of O 2 is lower than that of O (see figure 10 above).

Conclusion
A 2d3v PIC/MCC model was developed to investigate the physical processes in a magnetron discharge during the reactive sputter deposition of TiO x films. The plasma species taken into account are electrons, Ar + ions, fast Ar f atoms, metastable Ar * m atoms, Ti + ions, Ti atoms, O + ions, O + 2 ions, O − ions and O atoms. This model includes target sputtering, secondary electron emission, electron reflection, atom sticking and the effects of target poisoning (on sputter yield and SEEC value). To investigate the deposition process of the TiO x film, an analytical surface model was coupled to the PIC/MCC model. With this combined model, the cathode current and voltage, the plasma potential and electric field distribution, as well as the density profiles and fluxes of the different plasma species are calculated. Also, the stoichiometry x of the TiO x film and its deposition rate can be obtained in a self-consistent manner.
The plasma characteristics were calculated at different O 2 /Ar gas proportions. Note that at the lowest O 2 /Ar gas proportion, the target is already completely poisoned. It was shown that at constant Ar partial pressure, higher O 2 pressures cause higher O + and O + 2 densities. This leads to enhanced recombination reactions of these ions with the negative O − ions, causing their density to become smaller, and more spread out. This also results in a more spread out positive plasma potential toward the center of the reactor.
The deposition of the TiO x film was investigated, and it was calculated that at the O 2 /Ar gas proportions under study, the stoichiometry of x = 2 is almost achieved. When increasing the O 2 /Ar gas proportion, the stoichiometry rises to exactly x = 2, whereas the deposition rates of both Ti and O are hardly affected. The deposited O in the TiO x film originates almost entirely from the O 2 gas: O 2 molecules contribute at least 50 times more to the total oxygen deposition than O atoms.