Particle-in-cell/Monte Carlo collisions treatment of an Ar/O2 magnetron discharge used for the reactive sputter deposition of TiOx films
Identifieur interne : 001A95 ( Istex/Corpus ); précédent : 001A94; suivant : 001A96Particle-in-cell/Monte Carlo collisions treatment of an Ar/O2 magnetron discharge used for the reactive sputter deposition of TiOx films
Auteurs : E. Bultinck ; A. BogaertsSource :
- New Journal of Physics [ 1367-2630 ] ; 2009.
Abstract
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 plasmatarget 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 x2 for nearly all the investigated O2/Ar proportions.
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DOI: 10.1088/1367-2630/11/10/103010
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Particle-in-cell/Monte Carlo collisions treatment of an Ar/O2 magnetron discharge used for the reactive sputter deposition of TiOx films</title><author><name sortKey="Bultinck, E" sort="Bultinck, E" uniqKey="Bultinck E" first="E" last="Bultinck">E. Bultinck</name><affiliation><mods:affiliation>Research group PLASMANT, Department of Chemistry, University of Antwerp, Universiteitsplein 1, 2610 Antwerp, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>Author to whom any correspondence should be addressed.</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: evi.bultinck@ua.ac.be</mods:affiliation></affiliation></author><author><name sortKey="Bogaerts, A" sort="Bogaerts, A" uniqKey="Bogaerts A" first="A" last="Bogaerts">A. 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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 plasmatarget 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 x2 for nearly all the investigated O2/Ar proportions.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>E Bultinck</name><affiliations><json:string>Research group PLASMANT, Department of Chemistry, University of Antwerp, Universiteitsplein 1, 2610 Antwerp, Belgium</json:string><json:string>Author to whom any correspondence should be addressed.</json:string><json:string>E-mail: evi.bultinck@ua.ac.be</json:string></affiliations></json:item><json:item><name>A Bogaerts</name><affiliations><json:string>Research group PLASMANT, Department of Chemistry, University of Antwerp, Universiteitsplein 1, 2610 Antwerp, Belgium</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>paper</json:string></originalGenre><abstract>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 plasmatarget 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. 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Phys.</title><idno type="pISSN">1367-2630</idno><idno type="eISSN">1367-2630</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2009"></date><biblScope unit="volume">11</biblScope><biblScope unit="issue">10</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="24">24</biblScope></imprint></monogr><idno type="istex">6477F71A585C909271BD04BF8C871356FBEB587B</idno><idno type="DOI">10.1088/1367-2630/11/10/103010</idno><idno type="articleID">319880</idno><idno type="articleNumber">103010</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2009</date></creation><langUsage><language ident="en">en</language></langUsage><abstract><p>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 plasmatarget 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 x2 for nearly all the investigated O2/Ar proportions.</p></abstract></profileDesc><revisionDesc><change when="2009">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><original>false</original><mimetype>text/plain</mimetype><extension>txt</extension><uri>https://api.istex.fr/document/6477F71A585C909271BD04BF8C871356FBEB587B/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus iop not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="ISO-8859-1"</istex:xmlDeclaration><istex:docType SYSTEM="http://ej.iop.org/dtd/iopv1_5_2.dtd" name="istex:docType"></istex:docType><istex:document><article artid="nj319880"><article-metadata><jnl-data jnlid="nj"><jnl-fullname>New Journal of Physics</jnl-fullname><jnl-abbreviation>New J. Phys.</jnl-abbreviation><jnl-shortname>NJP</jnl-shortname><jnl-issn>1367-2630</jnl-issn><jnl-coden>NJOPFM</jnl-coden><jnl-imprint>IOP Publishing</jnl-imprint><jnl-web-address>stacks.iop.org/NJP</jnl-web-address></jnl-data><volume-data><year-publication>2009</year-publication><volume-number>11</volume-number></volume-data><issue-data><issue-number>10</issue-number><coverdate>October 2009</coverdate></issue-data><article-data><article-type type="paper" sort="regular"></article-type><type-number type="paper" numbering="article" artnum="103010">103010</type-number><article-number>319880</article-number><first-page>1</first-page><last-page>24</last-page><length>24</length><pii></pii><doi>10.1088/1367-2630/11/10/103010</doi><copyright>IOP Publishing and Deutsche Physikalische Gesellschaft</copyright><ccc>1367-2630/09/103010+24$30.00</ccc><printed></printed></article-data></article-metadata><header><title-group><title>Particle-in-cell/Monte Carlo collisions treatment of an <inline-eqn><math-text><upright>Ar</upright>/<upright>O</upright><sub>2</sub></math-text></inline-eqn> magnetron discharge used for the reactive sputter deposition of <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> films</title><short-title><inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> of reactive sputter deposition of <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn></short-title><ej-title>Particle-in-cell/Monte Carlo collisions treatment of an Ar/O2 magnetron discharge used for the reactive sputter deposition of TiOx films</ej-title></title-group><author-group><author address="nj319880ad1" alt-address="nj319880aad1" email="nj319880ea1"><first-names>E</first-names><second-name>Bultinck</second-name></author><author address="nj319880ad1"><first-names>A</first-names><second-name>Bogaerts</second-name></author><short-author-list>E Bultinck and A Bogaerts</short-author-list></author-group><address-group><address id="nj319880ad1" showid="no">Research group PLASMANT, Department of Chemistry, <orgname>University of Antwerp</orgname>, Universiteitsplein 1, 2610 Antwerp, <country>Belgium</country></address><address id="nj319880aad1" alt="yes" showid="yes">Author to whom any correspondence should be addressed.</address><e-address id="nj319880ea1"><email mailto="evi.bultinck@ua.ac.be">evi.bultinck@ua.ac.be</email></e-address></address-group><history received="26 May 2009" online="5 October 2009"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">The physical processes in an <inline-eqn><math-text><upright>Ar</upright>/<upright>O</upright><sub>2</sub></math-text></inline-eqn> magnetron discharge used for the reactive sputter deposition of <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> thin films were simulated with a 2d3v particle-in-<inline-eqn><math-text><upright>cell</upright>/<upright>Monte</upright></math-text></inline-eqn> Carlo collisions (<inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn>) model. The plasma species taken into account are electrons, <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions, fast <inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub></math-text></inline-eqn> atoms, metastable <inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn> atoms, <inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn> ions, Ti atoms, <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> 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 <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> gas ratio on the plasma potential and on the species densities and fluxes is investigated. Among others, it is shown that a higher <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> pressure causes the region of positive plasma potential and the <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> 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 <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> proportion. Indeed, the predicted stoichiometry of the deposited <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film approaches <inline-eqn><math-text><italic>x</italic>=2</math-text></inline-eqn> for nearly all the investigated <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> proportions.</p></abstract></abstract-group></header><body refstyle="numeric"><sec-level1 id="nj319880s1" label="1"><heading>Introduction</heading><p indent="no">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.</p><p>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:
<display-eqn id="nj319880eqn1" textype="equation" notation="LaTeX" eqnnum="1"></display-eqn>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 [<cite linkend="nj319880bib1">1</cite>]) 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.</p><p>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 <figref linkend="nj319880fig1">1</figref>). Therefore, the planar target is locally sputtered, leaving behind an erosion cavity, the so-called ‘race-track’.</p><figure id="nj319880fig1" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="17.3pc" printcolour="no" filename="images/nj319880fig1.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj319880fig1.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc1" label="Figure 1"><p indent="no">Schematic representation of the reactive sputter deposition process. O atoms, originating from plasma reactions with the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas (a), or from sputtering the poisoned <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> target (b), as well as <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas molecules, can react with sputtered Ti atoms on the substrate to form a <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film.</p></caption></figure><p>When a reactive gas, like oxygen, is added to the Ar background gas, the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas molecules themselves as well as atoms originating from the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas (by plasma reactions, or by sputtering a poisoned <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> target) can react with the sputtered metal atoms on the substrate to form a metal oxide layer [<cite linkend="nj319880bib2">2</cite>]–[<cite linkend="nj319880bib4">4</cite>], in a process called ‘reactive sputter deposition’. This process is schematically presented in figure <figref linkend="nj319880fig1">1</figref>.</p><p>However, these reactions between oxide and Ti species can also take place on the sputtering target, causing it to become (partially) oxidized. This target modification process is called ‘poisoning’. One major disadvantage of poisoning is that it may drastically reduce the sputtering yield of the target, and consequently the deposition rate [<cite linkend="nj319880bib5">5</cite>]. Furthermore, the transition from the so-called ‘metallic’ to ‘reactive’ condition happens via a complex and inconvenient hysteresis [<cite linkend="nj319880bib6">6</cite>, <cite linkend="nj319880bib7">7</cite>]. However, in order to deposit a stoichiometric <inline-eqn><math-text><upright>TiO</upright><sub>2</sub></math-text></inline-eqn> film, the reactive gas is necessary. The goal is to find reactive gas/background gas proportions which allow a satisfactory high deposition rate, as well as a stoichiometric deposited film.</p><p>Magnetron reactive sputter deposition is studied extensively by experiments (an overview is presented in [<cite linkend="nj319880bib8">8</cite>]). These experiments enable to discover and explain relations between external parameters, such as cathode current and voltage, gas pressure and pumping speed, magnetic field, geometrical characteristics of the magnetron, and the measured properties, such as current, voltage, pressure (if they change during the time progression of the discharge), plasma species densities, fluxes and energies, plasma potential distribution, erosion profiles and characterization of the deposited thin film. Some of these characteristics are however more precarious to measure due to possible plasma disturbance with the measuring tool and/or limitations in size of the measuring tool (hence certain areas are not reachable), or cannot be measured at all (for instance information on separate collision processes). However, to validate the developed numerical models, experiments remain inevitable. Moreover, certain input parameters are required in models, which need to be measured experimentally. Therefore, the combination of experiments and numerical models is desirable to provide a complete chemical and physical picture of magnetron discharges, needed to optimize the applications.</p><p>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.</p><p>‘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 [<cite linkend="nj319880bib9">9</cite>]–[<cite linkend="nj319880bib15">15</cite>]. The reactive sputter deposition process is modeled in [<cite linkend="nj319880bib5">5</cite>, <cite linkend="nj319880bib16">16</cite>, <cite linkend="nj319880bib17">17</cite>], by means of balance equations of the particle fluxes toward these surfaces.</p><p>‘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 [<cite linkend="nj319880bib18">18</cite>]. Nevertheless, in some papers [<cite linkend="nj319880bib19">19</cite>, <cite linkend="nj319880bib20">20</cite>] magnetron discharges are simulated by fluid models.</p><p>‘Hybrid models’ denote mostly the combination of Monte Carlo (MC) or particle-in-cell model/Monte Carlo collisions (<inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn>) 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.</p><p>‘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 [<cite linkend="nj319880bib21">21</cite>], 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 [<cite linkend="nj319880bib22">22</cite>]–[<cite linkend="nj319880bib24">24</cite>], which calculates the electric field in a self-consistent manner. The coupled model is named PIC/MCC model [<cite linkend="nj319880bib23">23</cite>, <cite linkend="nj319880bib24">24</cite>], 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.</p><p>Magnetron sputter deposition is investigated by MCC models [<cite linkend="nj319880bib25">25</cite>], and more precisely, the reactive sputter deposition was studied in [<cite linkend="nj319880bib26">26</cite>]. PIC/MCC models for sputter deposition in a pure Ar discharge are extensively developed [<cite linkend="nj319880bib23">23</cite>, <cite linkend="nj319880bib24">24</cite>], [<cite linkend="nj319880bib27">27</cite>]–[<cite linkend="nj319880bib32">32</cite>]. To the authors' knowledge, one PIC/MCC model exists for an Ar/O<inline-eqn><math-text><sub>2</sub></math-text></inline-eqn> gas mixture [<cite linkend="nj319880bib33">33</cite>]. However, this <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> 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 [<cite linkend="nj319880bib6">6</cite>, <cite linkend="nj319880bib7">7</cite>]. Secondly, the external circuit is not included in the model of [<cite linkend="nj319880bib33">33</cite>]. Nevertheless, the external circuit occurs to be inevitable in a <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> code for an accurate and correct description of magnetron discharges [<cite linkend="nj319880bib34">34</cite>].</p><p>In our opinion, the <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> modeling approach produces the most accurate data, making it a very powerful tool to simulate magnetron discharges. However, a complete <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> model does not yet exist to describe the reactive sputter deposition in an <inline-eqn><math-text><upright>Ar</upright>/<upright>O</upright><sub>2</sub></math-text></inline-eqn> gas mixture in a magnetron discharge. Therefore, to study the reactive magnetron sputter deposition process of <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> layers, a <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> model including plasma–surface interactions is developed and presented in this paper.</p><p>In the present work, the effect of the <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> 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 <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> proportions, as well as Ti, O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> fluxes to the substrate. The coupled analytical surface model enables to describe the deposition process of the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film, in terms of the deposited <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> fraction, the Ti and O deposition rates and the film stoichiometry <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>.</p></sec-level1><sec-level1 id="nj319880s2" label="2"><heading>Description of the model</heading><p indent="no">In the <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> 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 <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> model can be found in [<cite linkend="nj319880bib18">18</cite>], [<cite linkend="nj319880bib22">22</cite>]–[<cite linkend="nj319880bib24">24</cite>], [<cite linkend="nj319880bib32">32</cite>, <cite linkend="nj319880bib34">34</cite>]. A detailed description of a similar model developed for an <inline-eqn><math-text><upright>Ar</upright>/<upright>N</upright><sub>2</sub></math-text></inline-eqn> discharge can be found in [<cite linkend="nj319880bib35">35</cite>].</p><p>The species present in the discharge (electrons, <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions, fast <inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub></math-text></inline-eqn> atoms, metastable <inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn> atoms, <inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn> ions, Ti atoms, <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> 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 <inline-eqn><math-text><italic>V</italic></math-text></inline-eqn> is calculated from the densities of the negative species (electrons and negative ions) <inline-eqn><math-text><italic>n</italic><sub>−</sub></math-text></inline-eqn> and the densities of the positive ions <inline-eqn><math-text><italic>n</italic><sub>+</sub></math-text></inline-eqn>:
<display-eqn id="nj319880eqn2" textype="equation" notation="LaTeX" eqnnum="2"></display-eqn>This equation is solved with the superposition principle, presenting the potential <inline-eqn><math-text><italic>V</italic></math-text></inline-eqn> as the sum of the potential only due to the space charge, <inline-eqn><math-text><italic>V</italic><sub><upright>P</upright></sub></math-text></inline-eqn>, and the potential only due to the cathode voltage, <inline-eqn><math-text><italic>U</italic><sub>0</sub></math-text></inline-eqn>, i.e. <inline-eqn><math-text><italic>V</italic>=<italic>V</italic><sub><upright>P</upright></sub>+<italic>U</italic><sub>0</sub><italic>V</italic><sub><upright>L</upright></sub></math-text></inline-eqn>. <inline-eqn><math-text><italic>U</italic><sub>0</sub></math-text></inline-eqn> is calculated from the coupling of the external circuit to the plasma, and <inline-eqn><math-text><italic>V</italic><sub><upright>L</upright></sub></math-text></inline-eqn> is the dimensionless potential caused by an applied voltage with magnitude 1 V. A detailed description can be found in [<cite linkend="nj319880bib34">34</cite>]. The new electric field distribution <inline-eqn><math-text><italic>E</italic></math-text></inline-eqn> is calculated from <inline-eqn><math-text><italic>V</italic></math-text></inline-eqn>:
<display-eqn id="nj319880eqn3" textype="equation" notation="LaTeX" eqnnum="3"></display-eqn>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:
<display-eqn id="nj319880eqn4" textype="equation" notation="LaTeX" eqnnum="4"></display-eqn>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.</p><p>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 (<inline-eqn><math-text><italic>v</italic></math-text></inline-eqn>), the density of the target species (<inline-eqn><math-text><italic>n</italic><sub><upright>tar</upright></sub></math-text></inline-eqn>) and a collision probability, which is based on the collision cross section (<inline-eqn><math-text>σ</math-text></inline-eqn>), or on the reaction rate constant (<inline-eqn><math-text><italic>k</italic></math-text></inline-eqn>). The collision probability for collision <inline-eqn><math-text><italic>j</italic></math-text></inline-eqn> is presented as
<display-eqn id="nj319880eqn5" textype="equation" notation="LaTeX" eqnnum="5"></display-eqn>Some cross sections of ion–neutral collisions are described with the Langevin–Hasse model [<cite linkend="nj319880bib36">36</cite>, <cite linkend="nj319880bib37">37</cite>], assigned ‘LH’ in table <tabref linkend="nj319880tab1">1</tabref>, 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 [<cite linkend="nj319880bib38">38</cite>] (assumed 3 for all species [<cite linkend="nj319880bib37">37</cite>]), and on the masses, velocities and polarizability of the colliding species (11.08 <inline-eqn><math-text><italic>a</italic><sup>3</sup><sub>0</sub></math-text></inline-eqn> for Ar [<cite linkend="nj319880bib39">39</cite>], 10.60 <inline-eqn><math-text><italic>a</italic><sup>3</sup><sub>0</sub></math-text></inline-eqn> for <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> [<cite linkend="nj319880bib39">39</cite>] and 5.4 <inline-eqn><math-text><italic>a</italic><sup>3</sup><sub>0</sub></math-text></inline-eqn> for O [<cite linkend="nj319880bib39">39</cite>], where <inline-eqn><math-text><italic>a</italic><sub>0</sub></math-text></inline-eqn> is the Bohr radius).</p><table id="nj319880tab1" frame="topbot" position="float" width="page" place="top"><caption type="table" id="nj319880tc1" label="Table 1"><p>List of the collisions considered in the model. The references where the cross sections (<inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn>) or rate constants (<inline-eqn><math-text><italic>k</italic></math-text></inline-eqn>) were adopted from are presented in the last column. The rate constants are in <inline-eqn><math-text><upright>m</upright><sup>3</sup> <upright>s</upright><sup>−1</sup></math-text></inline-eqn> for two-body collisions and <inline-eqn><math-text><upright>m</upright><sup>6</sup> <upright>s</upright><sup>−1</sup></math-text></inline-eqn> for three body collisions (indicated by<inline-eqn><math-text><sup>*</sup></math-text></inline-eqn>). ‘(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 <secref linkend="nj319880s2">2</secref>).</p></caption><tgroup cols="5"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><colspec colnum="4" colname="col4" align="left"></colspec><colspec colnum="5" colname="col5" align="left"></colspec><tbody><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>e</upright><sup>−</sup></math-text></inline-eqn> collisions</entry></row><row><entry>(1)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>Ar</upright>→<upright>e</upright><sup>−</sup>+<upright>Ar</upright></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib53">53</cite>]</entry></row><row><entry>(2)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>Ar</upright>→2<upright>e</upright><sup>−</sup>+<upright>Ar</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Electron-impact ionization</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib54">54</cite>]</entry></row><row><entry>(3)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>Ar</upright>→<upright>e</upright><sup>−</sup>+<upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn></entry><entry>Electron-impact excitation</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib55">55</cite>]</entry></row><row><entry>(4)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>Ar</upright>→<upright>e</upright><sup>−</sup>+<upright>Ar</upright><sup>*</sup></math-text></inline-eqn></entry><entry>Electron-impact excitation</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib56">56</cite>]</entry></row><row><entry>(5)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup>→2<upright>e</upright><sup>−</sup>+<upright>Ar</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Electron-impact ionization</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib57">57</cite>]</entry></row><row><entry>(6)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup>→<upright>e</upright><sup>−</sup>+<upright>Ar</upright><sup>*</sup></math-text></inline-eqn></entry><entry>Electron-impact excitation</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib58">58</cite>]</entry></row><row><entry>(7)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>Ti</upright>→2<upright>e</upright><sup>−</sup>+<upright>Ti</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Electron-impact ionization</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib59">59</cite>]</entry></row><row><entry>(8)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→2<upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn></entry><entry>Electron-impact ionization</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib60">60</cite>, <cite linkend="nj319880bib61">61</cite>]</entry></row><row><entry>(9)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→<upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub><sup>*</sup>(<italic>a</italic>)</math-text></inline-eqn></entry><entry>Electron-impact excitation to <inline-eqn><math-text><italic>a</italic><sup>1</sup>&Dgr;<sub><italic>g</italic></sub></math-text></inline-eqn></entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib60">60</cite>]</entry></row><row><entry>(10)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→<upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub><sup>*</sup>(<italic>b</italic>)</math-text></inline-eqn></entry><entry>Electron-impact excitation to <inline-eqn><math-text><italic>b</italic><sup>1</sup>&Sgr;<sub><italic>g</italic></sub><sup>+</sup></math-text></inline-eqn></entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib60">60</cite>]</entry></row><row><entry>(11)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→<upright>O</upright><sup>−</sup>+<upright>O</upright></math-text></inline-eqn></entry><entry>Dissociative attachment</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib60">60</cite>]</entry></row><row><entry>(12)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→<upright>e</upright><sup>−</sup>+<upright>O</upright><sup>−</sup>+<upright>O</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Ion pair formation</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib33">33</cite>]</entry></row><row><entry>(13)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→2<upright>e</upright><sup>−</sup>+<upright>O</upright><sup>+</sup>+<upright>O</upright></math-text></inline-eqn></entry><entry>Dissociative ionization</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib60">60</cite>]</entry></row><row><entry>(14)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→<upright>e</upright><sup>−</sup>+<upright>O</upright>+<upright>O</upright></math-text></inline-eqn></entry><entry>Dissociation</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib60">60</cite>]</entry></row><row><entry>(15)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub><sup>+</sup>→<upright>O</upright>+<upright>O</upright></math-text></inline-eqn></entry><entry>Dissociative recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=2×10<sup>−13</sup>(300/<italic>T</italic><sub><upright>e</upright></sub>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib62">62</cite>, <cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(16)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>O</upright>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=6×10<sup>−39</sup>(300/<italic>T</italic><sub><upright>e</upright></sub>)<sup>1.5 *</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(17)</entry><entry><inline-eqn><math-text><upright>e</upright><sup>−</sup>+<upright>O</upright><sup>−</sup>→2<upright>e</upright><sup>−</sup>+<upright>O</upright></math-text></inline-eqn></entry><entry>Neutralization</entry><entry><inline-eqn><math-text><italic>k</italic>=1.95×10<sup>−18</sup>(<italic>T</italic><sub><upright>gas</upright></sub>/298)<sup>0.6</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib64">64</cite>]</entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> collisions</entry></row><row><entry>(18)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>Ar</upright>→<upright>Ar</upright><sup>+</sup>+<upright>Ar</upright></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib65">65</cite>]</entry></row><row><entry>(19)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>Ar</upright>→<upright>Ar</upright>+<upright>Ar</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib65">65</cite>]</entry></row><row><entry>(20)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>Ar</upright>→2<upright>Ar</upright><sup>+</sup>+<upright>e</upright><sup>−</sup></math-text></inline-eqn></entry><entry>Ion-impact ionization</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib66">66</cite>]</entry></row><row><entry>(21)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>Ar</upright>→<upright>Ar</upright><sup>+</sup>+<upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn></entry><entry>Ion-impact excitation</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib66">66</cite>]</entry></row><row><entry>(22)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>Ti</upright>→<upright>Ar</upright>+<upright>Ti</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=6.61×10<sup>−17</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib67">67</cite>]</entry></row><row><entry>(23)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>Ar</upright><sup>+</sup>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn> LH</entry><entry>[<cite linkend="nj319880bib33">33</cite>]</entry></row><row><entry>(24)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>Ar</upright>+<upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=5.1×10<sup>−17</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib68">68</cite>]</entry></row><row><entry>(25)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>O</upright>→<upright>Ar</upright><sup>+</sup>+<upright>O</upright></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn> LH</entry></row><row><entry>(26)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sup>+</sup>+<upright>O</upright>→<upright>Ar</upright>+<upright>O</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=6.4×10<sup>−18</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib68">68</cite>]</entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn> collisions</entry></row><row><entry>(27)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup>+<upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup>→<upright>Ar</upright>+<upright>Ar</upright><sup>+</sup>+<upright>e</upright><sup>−</sup></math-text></inline-eqn></entry><entry>Metastable-metastable collision</entry><entry><inline-eqn><math-text><italic>k</italic>=6.4×10<sup>−16</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib69">69</cite>, <cite linkend="nj319880bib70">70</cite>]</entry></row><row><entry>(28)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup>+<upright>Ti</upright>→<upright>Ar</upright>+<upright>Ti</upright><sup>+</sup>+<upright>e</upright><sup>−</sup></math-text></inline-eqn></entry><entry>Penning ionization</entry><entry><inline-eqn><math-text>σ=4.93×10<sup>−19</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib71">71</cite>]</entry></row><row><entry>(29)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup>+<upright>Ar</upright>→<upright>Ar</upright>+<upright>Ar</upright></math-text></inline-eqn></entry><entry>Two-body collision</entry><entry><inline-eqn><math-text><italic>k</italic>=2.3×10<sup>−21</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib72">72</cite>]</entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub></math-text></inline-eqn> collisions</entry></row><row><entry>(30)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub>+<upright>Ar</upright>→<upright>Ar</upright><sub>(<upright>f</upright>)</sub>+<upright>Ar</upright><sub>(<upright>f</upright>)</sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib73">73</cite>]</entry></row><row><entry>(31)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub>+<upright>Ar</upright>→<upright>Ar</upright>+<upright>Ar</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Atom-impact ionization</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib66">66</cite>]</entry></row><row><entry>(32)</entry><entry><inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub>+<upright>Ar</upright>→<upright>Ar</upright><sub><upright>f</upright></sub>+<upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn></entry><entry>Atom-impact excitation</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib74">74</cite>]</entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn> collisions</entry></row><row><entry>(33)</entry><entry><inline-eqn><math-text><upright>Ti</upright><sup>+</sup>+<upright>Ar</upright>→<upright>Ti</upright><sup>+</sup>+<upright>Ar</upright></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ=6×10<sup>−20</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib75">75</cite>]</entry></row><row><entry>(34)</entry><entry><inline-eqn><math-text><upright>Ti</upright><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>Ti</upright><sup>+</sup>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>LH</entry></row><row><entry>(35)</entry><entry><inline-eqn><math-text><upright>Ti</upright><sup>+</sup>+<upright>O</upright>→<upright>Ti</upright><sup>+</sup>+<upright>O</upright></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>LH</entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>Ti</upright></math-text></inline-eqn> collisions</entry></row><row><entry>(36)</entry><entry><inline-eqn><math-text><upright>Ti</upright><sub><upright>f</upright></sub>+<upright>Ar</upright>→<upright>Ti</upright>+<upright>Ar</upright><sub><upright>f</upright></sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ=6×10<sup>−20</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib75">75</cite>]</entry></row><row><entry>(37)</entry><entry><inline-eqn><math-text><upright>Ti</upright>+2<upright>O</upright>→<upright>TiO</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Attachment</entry><entry>Only at the walls (SC)</entry><entry></entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> collisions</entry></row><row><entry>(38)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>+</sup>+<upright>Ar</upright>→<upright>O</upright><sup>+</sup>+<upright>Ar</upright><sub><upright>f</upright></sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn> LH</entry><entry>[<cite linkend="nj319880bib33">33</cite>]</entry></row><row><entry>(39)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>+</sup>+<upright>Ar</upright>→<upright>O</upright>+<upright>Ar</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=3×10<sup>−17</sup></math-text></inline-eqn></entry><entry>(a)</entry></row><row><entry>(40)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>O</upright><sup>+</sup>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn> LH</entry><entry>[<cite linkend="nj319880bib33">33</cite>]</entry></row><row><entry>(41)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>O</upright>+<upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=3.3×10<sup>−17</sup><italic>e</italic><sup>−0.00169.T<sub><upright>gas</upright></sub></sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib62">62</cite>, <cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(42)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>+</sup>+<upright>O</upright><sub>3</sub>→<upright>O</upright><sub>2</sub><sup>+</sup>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Atom transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=1×10<sup>−16</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> collisions</entry></row><row><entry>(43)</entry><entry><inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup>+<upright>Ar</upright>→<upright>O</upright><sub>2</sub><sup>+</sup>+<upright>Ar</upright></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>LH</entry></row><row><entry>(44)</entry><entry><inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup>+<upright>Ar</upright>→<upright>O</upright><sub>2</sub>+<upright>Ar</upright><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=3×10<sup>−17</sup></math-text></inline-eqn></entry><entry>(a)</entry></row><row><entry>(45)</entry><entry><inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>O</upright><sub>2</sub><sup>+</sup>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn></entry><entry>LH</entry></row><row><entry>(46)</entry><entry><inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup>+<upright>O</upright><sub>2</sub>→<upright>O</upright><sub>2</sub>+<upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn></entry><entry>Charge transfer</entry><entry><inline-eqn><math-text><italic>k</italic>=3×10<sup>−17</sup></math-text></inline-eqn></entry><entry>(a)</entry></row><row><entry namest="col1" nameend="col5" align="left"><inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> collisions</entry></row><row><entry>(47)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>Ar</upright>→<upright>O</upright><sup>−</sup>+<upright>Ar</upright></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn> LH</entry><entry>[<cite linkend="nj319880bib33">33</cite>]</entry></row><row><entry>(48)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→<upright>O</upright><sup>−</sup>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Elastic scattering</entry><entry><inline-eqn><math-text>σ(<italic>E</italic>)</math-text></inline-eqn> LH</entry><entry>[<cite linkend="nj319880bib33">33</cite>]</entry></row><row><entry>(49)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>→<upright>e</upright><sup>−</sup>+<upright>O</upright><sub>3</sub></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=5×10<sup>−21</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(50)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>(<italic>a</italic>)→<upright>e</upright><sup>−</sup>+<upright>O</upright><sub>3</sub></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=3×10<sup>−16</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib62">62</cite>, <cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(51)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright><sub>2</sub>(<italic>b</italic>)→<upright>e</upright><sup>−</sup>+<upright>O</upright>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=6.9×10<sup>−16</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib62">62</cite>, <cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(52)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright>→<upright>e</upright><sup>−</sup>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=5×10<sup>−16</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(53)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright><sup>+</sup>→<upright>O</upright>+<upright>O</upright></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=2×10<sup>−13</sup>(300/<italic>T</italic><sub><upright>gas</upright></sub>)<sup>0.5</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(54)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright><sub>2</sub><sup>+</sup>→<upright>O</upright>+<upright>O</upright>+<upright>O</upright></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=1×10<sup>−13</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib63">63</cite>]</entry></row><row><entry>(55)</entry><entry><inline-eqn><math-text><upright>O</upright><sup>−</sup>+<upright>O</upright><sub>2</sub><sup>+</sup>→<upright>O</upright>+<upright>O</upright><sub>2</sub></math-text></inline-eqn></entry><entry>Recombination</entry><entry><inline-eqn><math-text><italic>k</italic>=2×10<sup>−13</sup>(300/<italic>T</italic><sub><upright>gas</upright></sub>)<sup>0.5</sup></math-text></inline-eqn></entry><entry>[<cite linkend="nj319880bib63">63</cite>]</entry></row></tbody></tgroup></table><p>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 [<cite linkend="nj319880bib40">40</cite>], 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 [<cite linkend="nj319880bib23">23</cite>, <cite linkend="nj319880bib32">32</cite>]. The collisions included in the model are summarized in table <tabref linkend="nj319880tab1">1</tabref>. Note that certain species that are present in the discharge are not followed in the model, since they are not very relevant, such as <inline-eqn><math-text><upright>O</upright><sub>3</sub></math-text></inline-eqn> (which is created by reactions (49) and (50)).</p><p>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 <italic>et al</italic> [<cite linkend="nj319880bib41">41</cite>]. 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, albeit, as neutral particles. Moreover, the metastable <inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn> atoms are reflected as ground state Ar atoms. In other words, the reactor walls act as a loss mechanism for <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn> metastable atoms, <inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> ions and <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> ions. In contrast to the above-mentioned species, Ti and O atoms, and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> 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 [<cite linkend="nj319880bib42">42</cite>], 1 for O [<cite linkend="nj319880bib5">5</cite>] and 0 for <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>. However, to describe the deposition process of the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film on the substrate, and to predict its stoichiometry, some effective sticking coefficients, <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn>, of Ti, O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> need to be calculated. These <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn> values are not constant, but are influenced by the incorporation or coverage of the different implanted or adsorbed species on the surface, respectively [<cite linkend="nj319880bib5">5</cite>]. Therefore, an analytical surface model, based on the models described in [<cite linkend="nj319880bib16">16</cite>, <cite linkend="nj319880bib17">17</cite>, <cite linkend="nj319880bib43">43</cite>] is coupled to the <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> model, in order to account for this effect of changing SC upon surface coverage by compound material. The general compound fraction balance equation for the deposition of <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> (giving rise to two deposited O species), O and Ti on the substrate to form a <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film with a desired stoichiometry <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> is [<cite linkend="nj319880bib16">16</cite>, <cite linkend="nj319880bib17">17</cite>, <cite linkend="nj319880bib43">43</cite>]
<display-eqn id="nj319880eqn6" textype="equation" notation="LaTeX" eqnnum="6"></display-eqn>Note that we use the static description of the analytical surface model. This means that <inline-eqn><math-text>&thetas;<sub><upright>cs</upright></sub><sup><italic>t</italic>, <italic>r</italic></sup></math-text></inline-eqn> is not calculated with its value from the previous time step, i.e. <inline-eqn><math-text><upright>d</upright>&thetas;<sub><upright>cs</upright></sub><sup><italic>t</italic>, <italic>r</italic></sup>/<upright>d</upright><italic>t</italic>=0</math-text></inline-eqn>. The time dependence arises from the PIC fluxes, <inline-eqn><math-text><italic>F</italic><sup><italic>t</italic>, <italic>r</italic></sup></math-text></inline-eqn>, which are calculated as a function of time.</p><p>Since the desired stoichiometry <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> of the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film is two, the surface model needs to account for this. Therefore, the compound fraction balance equation, for the deposition of a <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film with stoichiometry <inline-eqn><math-text><italic>x</italic>=2</math-text></inline-eqn>, is
<display-eqn id="nj319880eqn7" textype="equation" notation="LaTeX" eqnnum="7"></display-eqn>With this equation, at every <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> time step (<inline-eqn><math-text><italic>t</italic></math-text></inline-eqn>), the compound (<inline-eqn><math-text><italic>c</italic></math-text></inline-eqn>) fraction on every radial position (<inline-eqn><math-text><italic>r</italic></math-text></inline-eqn>) on the substrate (<inline-eqn><math-text><italic>s</italic></math-text></inline-eqn>), i.e. <inline-eqn><math-text>&thetas;<sub><upright>cs</upright></sub><sup><italic>t</italic>, <italic>r</italic></sup></math-text></inline-eqn>, is derived from the fluxes (<inline-eqn><math-text><italic>F</italic></math-text></inline-eqn>) of the incoming <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>, O and Ti species (calculated with the <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> model), with corresponding constant SC, which are chosen as 0.3 for <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> [<cite linkend="nj319880bib5">5</cite>], 1 for O [<cite linkend="nj319880bib5">5</cite>] and 0.5 for Ti [<cite linkend="nj319880bib42">42</cite>]:
<display-eqn id="nj319880eqn8" textype="equation" notation="LaTeX" eqnnum="8"></display-eqn>Subsequently, the SC values of O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> are adapted at that time step by the compound fraction, <inline-eqn><math-text>&thetas;<sub><upright>cs</upright></sub><sup><italic>t</italic>, <italic>r</italic></sup></math-text></inline-eqn>, so that we obtain the so-called effective sticking coefficients, <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn>, for this time step, on every position <inline-eqn><math-text><italic>r</italic></math-text></inline-eqn> on the substrate:
<display-eqn id="nj319880eqn9" textype="equation" notation="LaTeX" eqnnum="9"></display-eqn>Indeed, the sticking of O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> on <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> 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 <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> surface), a constant <inline-eqn><math-text><upright>SC</upright><sub><upright>Ti</upright></sub></math-text></inline-eqn> can be used, i.e. <inline-eqn><math-text><upright>SC</upright><sub><upright>Ti</upright>, <upright>eff</upright></sub><sup><italic>t</italic>, <italic>r</italic></sup>=<upright>SC</upright><sub><upright>Ti</upright></sub></math-text></inline-eqn>. Note that the actual stoichiometry of the deposited <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film at steady state, <inline-eqn><math-text><italic>x</italic><sup><italic>t</italic>, <italic>r</italic></sup></math-text></inline-eqn>, is found by multiplying the desired stoichiometry, <inline-eqn><math-text><italic>x</italic>=2</math-text></inline-eqn>, with the compound fraction, <inline-eqn><math-text>&thetas;<sub><upright>cs</upright></sub><sup><italic>t</italic>, <italic>r</italic></sup></math-text></inline-eqn>.</p><p>When an SP is created or reflected at a wall, the direction of its velocity is chosen as such to obey the cosine law [<cite linkend="nj319880bib44">44</cite>]. As a consequence, the azimuthal angle, <inline-eqn><math-text>&phis;=2π.<upright>RN</upright>[0, 1]</math-text></inline-eqn>, whereas the polar angle, <inline-eqn></inline-eqn>. RN[0, 1] is a random number uniformly distributed in the interval <inline-eqn><math-text>[0, 1]</math-text></inline-eqn>. For emission of an electron from the cathode surface, the components of the velocity vector <inline-eqn><math-text><bold>v</bold></math-text></inline-eqn>, <inline-eqn><math-text><italic>v</italic><sub><italic>r</italic></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>&thetas;</sub></math-text></inline-eqn> are correspondent to the azimuthal angle, <inline-eqn><math-text>&phis;</math-text></inline-eqn>, whereas <inline-eqn><math-text><italic>v</italic><sub><italic>z</italic></sub></math-text></inline-eqn> is derived from the polar angle, <inline-eqn><math-text>&thetas;</math-text></inline-eqn>. In cylindrical coordinates, this becomes:
<display-eqn id="nj319880eqn10" textype="equation" notation="LaTeX" eqnnum="10" lines="multiline"></display-eqn>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, <inline-eqn><math-text>α</math-text></inline-eqn>:
<display-eqn id="nj319880eqn11" textype="equation" notation="LaTeX" eqnnum="11"></display-eqn>where <inline-eqn><math-text><italic>E</italic><sub><upright>i</upright></sub></math-text></inline-eqn> is the mean energy of the incoming particles, <inline-eqn><math-text><italic>E</italic><sub><upright>r</upright></sub></math-text></inline-eqn> is the mean energy of the reflected particles and <inline-eqn><math-text><italic>E</italic><sub><upright>w</upright></sub></math-text></inline-eqn> is the mean energy of the reflected particles in thermal equilibrium with the wall (<inline-eqn><math-text>2<italic>kT</italic><sub><upright>w</upright></sub></math-text></inline-eqn>). In our simulation, <inline-eqn><math-text>α</math-text></inline-eqn> is assumed as 0.5 [<cite linkend="nj319880bib45">45</cite>], and a random number between 0 and 1 is generated. When the random number is lower than <inline-eqn><math-text>α</math-text></inline-eqn>, an inelastic reflection occurs, and if the random number is higher than <inline-eqn><math-text>α</math-text></inline-eqn>, 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 [<cite linkend="nj319880bib44">44</cite>], and its components <inline-eqn><math-text>(<italic>v</italic><sub><italic>r</italic></sub>, <italic>v</italic><sub>&thetas;</sub>, <italic>v</italic><sub><italic>z</italic></sub>)</math-text></inline-eqn> are as in equation (<eqnref linkend="nj319880eqn10">10</eqnref>).</p><p>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 <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> 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 [<cite linkend="nj319880bib46">46</cite>]. In the model, the poisoning of the target is therefore described by changing the SEEC value, depending on the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas pressure, and in correspondence to the range reported in [<cite linkend="nj319880bib46">46</cite>]. 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 <italic>et al</italic> [<cite linkend="nj319880bib41">41</cite>], is lowered by a factor, in accordance to the values reported in [<cite linkend="nj319880bib47">47</cite>]: the sputter yield of Ti from a fully poisoned <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> 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 [<cite linkend="nj319880bib47">47</cite>].</p><p>When treating the slow (thermalized) particles with the <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> 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 <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>) are initially slow, but can be heated [<cite linkend="nj319880bib48">48</cite>]. The partial pressure of the gases, <inline-eqn><math-text><italic>p</italic><sub><upright>Ar</upright></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>p</italic><sub><upright>O</upright><sub>2</sub></sub></math-text></inline-eqn> 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, <inline-eqn><math-text><italic>T</italic><sub><upright>Ar</upright></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic><sub><upright>O</upright><sub>2</sub></sub></math-text></inline-eqn> will increase. The densities of the gases, <inline-eqn><math-text><italic>n</italic><sub><upright>Ar</upright></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>n</italic><sub><upright>O</upright><sub>2</sub></sub></math-text></inline-eqn>, which change due to gas heating, can be calculated at every position and at every time step, with the ideal gas law:
<display-eqn id="nj319880eqn12" textype="equation" notation="LaTeX" eqnnum="12"></display-eqn>Here, <inline-eqn><math-text><italic>k</italic><sub><upright>B</upright></sub></math-text></inline-eqn> is the Boltzmann constant. The flux of <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> is needed to picture the deposition of the TiO<inline-eqn><math-text><sub><italic>x</italic></sub></math-text></inline-eqn> film. The flux is calculated as the amount of collisions with a surface per second:
<display-eqn id="nj319880eqn13" textype="equation" notation="LaTeX" eqnnum="13"></display-eqn>The velocity, <inline-eqn><math-text><italic>v</italic></math-text></inline-eqn>, is the average velocity of the gas molecules, i.e. thermal velocity <inline-eqn></inline-eqn>. The diffusing atoms can be created as slow particles, or can be slowed down by collisions (Ti and O). Their density <inline-eqn><math-text><italic>n</italic><sub><upright>slow</upright></sub>(<italic>r</italic>, <italic>z</italic>)</math-text></inline-eqn> is calculated with the diffusion equation:
<display-eqn id="nj319880eqn14" textype="equation" notation="LaTeX" eqnnum="14"></display-eqn>where <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn> is the diffusion coefficient of Ti or O atoms in Ar gas, and <inline-eqn><math-text><italic>r</italic><sub><upright>prod</upright></sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>r</italic><sub><upright>loss</upright></sub></math-text></inline-eqn> are the production and loss rates. The production of a slow Ti or O atom (<inline-eqn><math-text><italic>r</italic><sub><upright>prod</upright></sub></math-text></inline-eqn>) 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 slow Ti or O atom (<inline-eqn><math-text><italic>r</italic><sub><upright>loss</upright></sub></math-text></inline-eqn>) can be due to heating of a slow Ti or O atom. The diffusion coefficients <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn> (in <inline-eqn><math-text><upright>m</upright><sup>2</sup> <upright>s</upright><sup>−1</sup></math-text></inline-eqn>) are calculated from the rigid sphere approximation [<cite linkend="nj319880bib49">49</cite>]:
<display-eqn id="nj319880eqn15" textype="equation" notation="LaTeX" eqnnum="15"></display-eqn>where <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn> is the gas temperature (K), <inline-eqn><math-text><italic>M</italic><sub>1</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>M</italic><sub>2</sub></math-text></inline-eqn> are the masses of Ti (or O) and Ar (g/mole), <inline-eqn><math-text><italic>p</italic></math-text></inline-eqn> is the pressure (atm) and <inline-eqn><math-text><italic>d</italic><sub>12</sub></math-text></inline-eqn> is the collision diameter (<inline-eqn><math-text>10<sup>−10</sup> <upright>m</upright></math-text></inline-eqn>), given by <inline-eqn></inline-eqn>. The collision diameter of Ti is <inline-eqn><math-text>2.684×10<sup>−10</sup> <upright>m</upright></math-text></inline-eqn> [<cite linkend="nj319880bib50">50</cite>], <inline-eqn><math-text>3.050×10<sup>−10</sup> <upright>m</upright></math-text></inline-eqn> for O [<cite linkend="nj319880bib51">51</cite>] and <inline-eqn><math-text>3.542×10<sup>−10</sup> <upright>m</upright></math-text></inline-eqn> for Ar [<cite linkend="nj319880bib51">51</cite>].</p></sec-level1><sec-level1 id="nj319880s3" label="3"><heading>Results and discussion</heading><sec-level2 id="nj319880s3.1" label="3.1"><heading>General conditions</heading><p indent="no">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 <figref linkend="nj319880fig2">2</figref>. 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 <figref linkend="nj319880fig2">2</figref>.</p><figure id="nj319880fig2" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="17.9pc" printcolour="no" filename="images/nj319880fig2.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj319880fig2.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc2" label="Figure 2"><p indent="no">Scheme of the simulated two-dimensional magnetron area, with the external electrical circuit generating an electric field, and the external magnets, creating a magnetic field (magnetic field vectors are indicated on the figure). The cylindrical three-dimensional magnetron discharge is apparent by rotating around the dashed symmetry axis.</p></caption></figure><p>An electrical circuit, consisting of an external resistance (<inline-eqn><math-text><italic>R</italic><sub><upright>ext</upright></sub>=1500 &OHgr;</math-text></inline-eqn>) and voltage source (<inline-eqn><math-text><italic>V</italic><sub><upright>ext</upright></sub>=−600 <italic>V</italic></math-text></inline-eqn>), is connected to the cathode, whereas the other walls are grounded, to generate an electric field. The coupling of the external circuit to the <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> code is explained in [<cite linkend="nj319880bib34">34</cite>].</p><p>In order to study the effect of the <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> gas ratio, the Ar partial pressure was kept constant at 1 Pa for all the calculations, while the partial pressure of <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> was varied as 0.02, 0.06, 0.12, 0.18 and 0.24 Pa, according to measured values. The calculated currents and voltages at the different <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> gas proportions are in the range of 0.18–0.2 A and 310–330 V, respectively. The values correspond well with experimental values at similar conditions [<cite linkend="nj319880bib52">52</cite>].</p></sec-level2><sec-level2 id="nj319880s3.2" label="3.2"><heading>Potential distribution, and charged species density profiles</heading><p indent="no">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 (<eqnref linkend="nj319880eqn2">2</eqnref>)). 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 <figref linkend="nj319880fig3">3</figref> (for the case of 0.12 Pa <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>). The electron density is only weakly dependent on the <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> proportion.</p><figure id="nj319880fig3" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="21.9pc" printcolour="no" filename="images/nj319880fig3.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj319880fig3.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc3" label="Figure 3"><p indent="no">Calculated electron density profile (in <inline-eqn><math-text><upright>m</upright><sup>−3</sup></math-text></inline-eqn>) at an Ar partial pressure of 1.0 Pa and an <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures of 0.12 Pa. Note that the <inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn>-axis (<inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn>) corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>The electrons ionize neutral Ar, <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>, 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 <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> density has approximately the same magnitude as the electron density, the <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> densities are two orders of magnitude lower (the latter is shown in figure <figref linkend="nj319880fig4">4</figref>), and the <inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn> density is three orders of magnitude lower. Both the <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> and the <inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn> density do not change much with <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> ratio. However, the <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> densities rise with <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas pressure, as is clear from figure <figref linkend="nj319880fig4">4</figref>. 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.</p><figure id="nj319880fig4" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="21.9pc" printcolour="no" filename="images/nj319880fig4.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj319880fig4.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc4" label="Figure 4"><p indent="no">Calculated <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> density profiles (in <inline-eqn><math-text><upright>m</upright><sup>−3</sup></math-text></inline-eqn>) at an Ar partial pressure of 1.0 Pa and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures of 0.02, 0.12 and 0.24 Pa. Note that the <inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn>-axis (<inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn>) corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>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 <figref linkend="nj319880fig5">5</figref>. 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 <figref linkend="nj319880fig5">5</figref>.</p><figure id="nj319880fig5" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="18.9pc" printcolour="no" filename="images/nj319880fig5.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj319880fig5.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc5" label="Figure 5"><p indent="no">Calculated plasma potential distribution at an Ar partial pressure of 1.0 Pa, without <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> and at <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures of 0.02, 0.12 and 0.24 Pa. Note that the <inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn>-axis (<inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn>) corresponds to the symmetry axis of the cylindrically symmetrical reactor. The line that separates the negative and positive potential area (<inline-eqn><math-text><italic>V</italic>=0</math-text></inline-eqn>) is marked thicker for clarity.</p></caption></figure><p>When no <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> 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 <figref linkend="nj319880fig5">5</figref> (top figure).</p><p>When <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> is added to the Ar gas, negative <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> ions are created by the electron impact reactions (<eqnref linkend="nj319880eqn11">11</eqnref>) and (<eqnref linkend="nj319880eqn12">12</eqnref>) from table <tabref linkend="nj319880tab1">1</tabref>. Because the electrons are confined to the magnetic field in an area close to the cathode, the <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> ions are also created in this area. However, taking into account that heavy <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> ions are not trapped in the magnetic field in contrast to electrons, the plasma potential distribution (figure <figref linkend="nj319880fig5">5</figref>) forces the negative <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> ions to move towards the positive potential area. Figure <figref linkend="nj319880fig6">6</figref> presents the calculated <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> density, which has a maximum at the positive potential area. This negative ion density causes the positive potential area to enlarge, see figure <figref linkend="nj319880fig5">5</figref> (compare ‘without <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>’ with ‘0.02 Pa <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>’).</p><figure id="nj319880fig6" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="21.9pc" printcolour="no" filename="images/nj319880fig6.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj319880fig6.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc6" label="Figure 6"><p indent="no">Calculated <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> density profiles (in <inline-eqn><math-text><upright>m</upright><sup>−3</sup></math-text></inline-eqn>) at an Ar partial pressure of 1.0 Pa and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures of 0.02, 0.12 and 0.24 Pa. Note that the <inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn>-axis (<inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn>) corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>Figure <figref linkend="nj319880fig5">5</figref> also illustrates that at increasing <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> density, the positive potential area moves toward the symmetry axis of the magnetron. This is also true for the <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> density (see figure <figref linkend="nj319880fig6">6</figref>), which also becomes smaller and more spread out. The decreasing trend of the <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> density is a bit unexpected, but it is a consequence of the <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> densities, which rise significantly when the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> pressure is increased, as was clear from figure <figref linkend="nj319880fig4">4</figref> above. Consequently, the recombination reactions (53), (54) and (55) from table <tabref linkend="nj319880tab1">1</tabref> occur much more often, which are an important loss mechanism for <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> ions. Therefore, the <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> density decreases upon <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> addition (see figure <figref linkend="nj319880fig6">6</figref>). When the density drops, the Debye length increases leading to a less pronounced <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> 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.</p></sec-level2><sec-level2 id="nj319880s3.3" label="3.3"><heading>Calculated sputtered Ti and O fluxes</heading><p indent="no">The plasma species can bombard the target surface and sputter Ti and O atoms. Positive <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn>, <inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn>, <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> ions, and fast <inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub></math-text></inline-eqn> atoms contribute to the sputtering of the poisoned <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> target. The sputtered Ti and O fluxes at different values of the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressure are presented in figure <figref linkend="nj319880fig7">7</figref>. 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 <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> 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.</p><figure id="nj319880fig7" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="26.0pc" printcolour="no" filename="images/nj319880fig7.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj319880fig7.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc7" label="Figure 7"><p indent="no">Calculated sputtered Ti and O fluxes from the cathode for different <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures as indicated in the legend, and at an Ar partial pressure of 1.0 Pa. Note that the <inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn>-axis (<inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn>) corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>The sputtered Ti and O fluxes are approximately equal, although the sputter yield of O is two times higher than of Ti (see section <secref linkend="nj319880s2">2</secref>). This is caused by the fact that the threshold energy for sputtering Ti by <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions is lower (the main sputter source in the discharge) than sputtering O by <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions (the threshold energy for sputtering Ti with <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions is 12.55 eV and 18.85 eV for sputtering O). Also, there is no dependence of the amount of sputtering on the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressure, since most sputtering is caused by <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions, which flux is independent of the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressure.</p></sec-level2><sec-level2 id="nj319880s3.4" label="3.4"><heading>Calculated Ti, O and O<sub>2</sub> fluxes to the substrate</heading><p indent="no">The sputtered Ti and O atoms, together with the produced O atoms by plasma reactions, as well as the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas molecules, move through the plasma and can be deposited on a substrate, placed in front of the target. The calculated Ti, O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> fluxes to the substrate are shown in figure <figref linkend="nj319880fig8">8</figref>. <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn>, <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> ions are also present close to the substrate (see figures <figref linkend="nj319880fig4">4</figref> and <figref linkend="nj319880fig6">6</figref>). However, their flux to the substrate is several orders of magnitude lower than the O flux, causing these ions to have a negligible contribution to the deposition of the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film.</p><figure id="nj319880fig8" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="17.0pc" printcolour="no" filename="images/nj319880fig8.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj319880fig8.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc8" label="Figure 8"><p indent="no">Calculated fluxes of the Ti, O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> species to the substrate, as a function of radial position, for different <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures, at an Ar partial pressure of 1 Pa. The value <inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn> corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>The Ti flux to the anode has a peak profile caused by the sputtered Ti peak profile (see figure <figref linkend="nj319880fig7">7</figref>). 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 <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> amount, in correspondence to the sputtered Ti flux (see section <secref linkend="nj319880s3.3">3.3</secref> and figure <figref linkend="nj319880fig7">7</figref> above).</p><p>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 <figref linkend="nj319880fig7">7</figref>), because more O atoms are created by plasma reactions than by sputtering. The O flux increases upon <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> addition, which is a consequence of higher O production at higher <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> amount.</p><p>At the gas pressures under consideration, gas heating is not so important [<cite linkend="nj319880bib48">48</cite>]. As a result the Ar and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas densities remain constant and uniform (see also equation (<eqnref linkend="nj319880eqn12">12</eqnref>)). Consequently, the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> flux bombarding the substrate has a uniform spatial profile. Logically, the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> flux increases for higher <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressure (see also equation (<eqnref linkend="nj319880eqn13">13</eqnref>)).</p></sec-level2><sec-level2 id="nj319880s3.5" label="3.5"><heading>Calculated deposition of a TiO<sub>x</sub> film on the substrate</heading><p indent="no">Based on these calculated Ti, O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> fluxes (figure <figref linkend="nj319880fig8">8</figref>) and the corresponding SC values, the radial deposited fraction of <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> (i.e. the ‘compound fraction’) on the substrate is calculated with equation (<eqnref linkend="nj319880eqn8">8</eqnref>), and it is presented in figure <figref linkend="nj319880fig9">9</figref>. It is clear that the compound fraction is close to unity for all <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> 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 (<eqnref linkend="nj319880eqn8">8</eqnref>). At increasing <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> amount, the compound fraction on the substrate increases as a result of the higher O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> fluxes, and it becomes more uniform because these oxygen fluxes gain importance in the compound coverage, and they have a more uniform profile compared to the peaking Ti flux.</p><figure id="nj319880fig9" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="17.9pc" printcolour="no" filename="images/nj319880fig9.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj319880fig9.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc9" label="Figure 9"><p indent="no">Calculated compound fraction on the substrate surface, as a function of radial position, for different <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures, at an Ar partial pressure of 1 Pa. The value <inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn> corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>Based on this radial compound fraction, the constant SC values of O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> are adapted, according to equation (<eqnref linkend="nj319880eqn9">9</eqnref>). The calculated effective SC values (<inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn>) of O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> are presented in figure <figref linkend="nj319880fig10">10</figref>. In contrast to the radially constant SC values of a metallic substrate, the effective <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn> values are not constant in the radial direction. Indeed, the <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn> profile exhibits a radial dependence, being lowest at the highest compound fraction. In comparison with the constant initial SC values of O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>, i.e. 1 and 0.3, the effective values have dropped by about a factor of 25–250. Moreover, the effective sticking of the O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> species decreases at increasing <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas pressure, caused by the higher compound fraction.</p><figure id="nj319880fig10" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="26.0pc" printcolour="no" filename="images/nj319880fig10.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj319880fig10.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc10" label="Figure 10"><p indent="no">Calculated effective sticking coefficients, <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn>, of O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn>, as a function of radial position, for different <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures, at an Ar partial pressure of 1 Pa. The value <inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn> corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>The deposition rate of Ti and O can be found when multiplying the fluxes of Ti, O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> (figure <figref linkend="nj319880fig8">8</figref>) by the corresponding <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn> values. The total deposition rate of oxygen is the sum of the deposition rates due to O and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> (note that <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> counts double, because one <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> molecule gives rise to two deposited O atoms). Figure <figref linkend="nj319880fig11">11</figref> shows that the total deposition rate of O is two times higher than the deposition rate of Ti, which is required to deposit a <inline-eqn><math-text><upright>TiO</upright><sub>2</sub></math-text></inline-eqn> film. The calculated stoichiometries for three of the investigated <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures are also presented (figure <figref linkend="nj319880fig11">11</figref>: gray curves, right axis). At the lowest <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> amount (0.02 Pa), the stoichiometry of two is almost achieved (with a minimum value of 1.93). At increasing <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> 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 <figref linkend="nj319880fig9">9</figref>: the real stoichiometry of the film is found by multiplying the desired stoichiometry <inline-eqn><math-text><italic>x</italic>=2</math-text></inline-eqn> by the compound coverage (note that the desired stoichiometry <inline-eqn><math-text><italic>x</italic>=2</math-text></inline-eqn> was imposed by equation (<eqnref linkend="nj319880eqn7">7</eqnref>)). This means that an <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> proportion of 1/10 is already enough to deposit a stoichiometric film (with a minimum value of 1.98), with a satisfactory deposition rate.</p><figure id="nj319880fig11" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="37.9pc" printcolour="no" filename="images/nj319880fig11.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj319880fig11.jpg"></graphic-file></graphic><caption type="figure" id="nj319880fc11" label="Figure 11"><p indent="no">Calculated deposition rate of Ti and total O as a function of radial position (black curves, left axis) at an Ar partial pressure of 1 Pa and <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> partial pressures of 0.02, 0.12 and 0.24 Pa. Also shown is the calculated stoichiometry <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> of the deposited TiO<inline-eqn><math-text><sub><italic>x</italic></sub></math-text></inline-eqn> films (gray curves, right axis). The value <inline-eqn><math-text><italic>r</italic>=0</math-text></inline-eqn> corresponds to the symmetry axis of the cylindrically symmetrical reactor.</p></caption></figure><p>The peak profiles in figure <figref linkend="nj319880fig11">11</figref> 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 <figref linkend="nj319880fig8">8</figref>) leading to radially dependent <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn> values (figure <figref linkend="nj319880fig10">10</figref>). 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 <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> contributes at least 50 times more than O to the total deposition of oxygen in the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film. This is caused by the much higher <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> flux compared to the O flux (see figure <figref linkend="nj319880fig8">8</figref> above), although the <inline-eqn><math-text><upright>SC</upright><sub><upright>eff</upright></sub></math-text></inline-eqn> of <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> is lower than that of O (see figure <figref linkend="nj319880fig10">10</figref> above).</p></sec-level2></sec-level1><sec-level1 id="nj319880s4" label="4"><heading>Conclusion</heading><p indent="no">A 2d3v <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> model was developed to investigate the physical processes in a magnetron discharge during the reactive sputter deposition of <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> films. The plasma species taken into account are electrons, <inline-eqn><math-text><upright>Ar</upright><sup>+</sup></math-text></inline-eqn> ions, fast <inline-eqn><math-text><upright>Ar</upright><sub><upright>f</upright></sub></math-text></inline-eqn> atoms, metastable <inline-eqn><math-text><upright>Ar</upright><sub><upright>m</upright></sub><sup>*</sup></math-text></inline-eqn> atoms, <inline-eqn><math-text><upright>Ti</upright><sup>+</sup></math-text></inline-eqn> ions, Ti atoms, <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> ions, <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> 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 <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film, an analytical surface model was coupled to the <inline-eqn><math-text><upright>PIC</upright>/<upright>MCC</upright></math-text></inline-eqn> 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 <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> of the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film and its deposition rate can be obtained in a self-consistent manner.</p><p>The plasma characteristics were calculated at different <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> gas proportions. Note that at the lowest <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> gas proportion, the target is already completely poisoned. It was shown that at constant Ar partial pressure, higher <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> pressures cause higher <inline-eqn><math-text><upright>O</upright><sup>+</sup></math-text></inline-eqn> and <inline-eqn><math-text><upright>O</upright><sub>2</sub><sup>+</sup></math-text></inline-eqn> densities. This leads to enhanced recombination reactions of these ions with the negative <inline-eqn><math-text><upright>O</upright><sup>−</sup></math-text></inline-eqn> 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.</p><p>The deposition of the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film was investigated, and it was calculated that at the <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> gas proportions under study, the stoichiometry of <inline-eqn><math-text><italic>x</italic>=2</math-text></inline-eqn> is almost achieved. When increasing the <inline-eqn><math-text><upright>O</upright><sub>2</sub>/<upright>Ar</upright></math-text></inline-eqn> gas proportion, the stoichiometry rises to exactly <inline-eqn><math-text><italic>x</italic>=2</math-text></inline-eqn>, whereas the deposition rates of both Ti and O are hardly affected. The deposited O in the <inline-eqn><math-text><upright>TiO</upright><sub><italic>x</italic></sub></math-text></inline-eqn> film originates almost entirely from the <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> gas: <inline-eqn><math-text><upright>O</upright><sub>2</sub></math-text></inline-eqn> molecules contribute at least 50 times more to the total oxygen deposition than O atoms.</p></sec-level1><acknowledgment><heading>Acknowledgments</heading><p indent="no">EB is indebted to the University of Antwerp for financial support. We thank S Mahieu and D Depla for providing the experimental data, and for the interesting discussions on the plasma–surface interactions, for which W Möller is also greatly acknowledged. 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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 plasmatarget 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 x2 for nearly all the investigated O2/Ar proportions.</abstract><relatedItem type="host"><titleInfo><title>New Journal of Physics</title></titleInfo><titleInfo type="abbreviated"><title>New J. Phys.</title></titleInfo><genre type="journal">journal</genre><identifier type="ISSN">1367-2630</identifier><identifier type="eISSN">1367-2630</identifier><identifier type="PublisherID">nj</identifier><identifier type="CODEN">NJOPFM</identifier><identifier type="URL">stacks.iop.org/NJP</identifier><part><date>2009</date><detail type="volume"><caption>vol.</caption><number>11</number></detail><detail type="issue"><caption>no.</caption><number>10</number></detail><extent unit="pages"><start>1</start><end>24</end><total>24</total></extent></part></relatedItem><identifier type="istex">6477F71A585C909271BD04BF8C871356FBEB587B</identifier><identifier type="DOI">10.1088/1367-2630/11/10/103010</identifier><identifier type="articleID">319880</identifier><identifier type="articleNumber">103010</identifier><accessCondition type="use and reproduction" contentType="copyright">IOP Publishing and Deutsche Physikalische Gesellschaft</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>IOP Publishing and Deutsche Physikalische Gesellschaft</recordOrigin></recordInfo></mods></metadata><enrichments><json:item><type>multicat</type><uri>https://api.istex.fr/document/6477F71A585C909271BD04BF8C871356FBEB587B/enrichments/multicat</uri></json:item></enrichments><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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