TheSputteringProblemJamesAGlackin,JamesV.MathesorI proposetoconsiderfirstthevarious elements ofthesubject,next itsvariousparts or sections,andfinallythewhole in its internalstructure.Inotherwords,Ishallproceedfromthesimpletothecomplex.CarlVonClausewitzIntroductionGuntubesproducedforArmyweaponssystems,suchastanksandartillerypieces,aredesignedto fire a specific number of rounds before they areconsideredunsafeandmustbereplaced.TheH·Kexact round countfora gun tube is so critical thatdetailed recordsarekeptforeachtube ineveryArmorandArtilleryunit in theArmy.Frequent replacement of gun tubes isundesirablenotonlyforobviousfiscalreasons,butalsoduetothemanpowerandfirepowerdrainitplacesonaunitincombat.Onemethodofincreasingthenumberofroundsaguntubecansafelyfireistoplaceametal lininginsidethetube.One of thetechniques forplacing a lining insidea gun tube is calledsputtering.Sputtering is a process where a cylindrical metal bar is inserted insidea guntube,centered,anchoredinplace,andconnectedtoapowersource.Thetubeandrodareplacedinsideachamberwhereallairisevacuatedandreplacedwitha noblegas.When electricityisrunthroughthesourcerod,amagneticfielddevelopsandatomsfromthemetalsourcerodarebrokenawayand literallyfly across thegap from the sourcerod to the inside ofthe gun tube.Duringflight, themetal atoms collide with atoms of the noble gas that separatethesource rod fromthe inside wall of the gun tube.Each collision with a noble gasatomcausesthemetal atomto changethedirectionof itsflight and loseenergy.Indevelopingthesputteringprocess,theArmyusedthreemathematicalmodelstocreatecomputersimulationsofthesputteringprocess.The computer simulations wereusedtoquickly and inexpensivelypredict theeffects of different materials and experimental conditions on the sputteringresults.One model was used to simulate initial direction and energy of asputtered atom.Asecondmodel was usedto analyzetheeffects of anatomcollidingwiththeinsideoftheguntubewall.Thischapterdealswiththethirdmathematical model, whichwas developedtoanalyzetheflightofametal atomfrom the source rod to the inside of thegun tube.Inordertomodelthisportionoftheproblem,it isnecessarytoanalyzetheflightofanatomfromthesourcerod (cathode)totheinsideoftheguntube (anode)in87
87 The Sputtering Problem James A Glackin, James V. Matheson I propose to consider first the various elements of the subject, next its various parts or sections, and finally the whole in its internal structure. In other words, I shall proceed from the simple to the complex. Carl Von Clausewitz Introduction Gun tubes produced for Army weapons systems, such as tanks and artillery pieces, are designed to fire a specific number of rounds before they are considered unsafe and must be replaced. The exact round count for a gun tube is so critical that detailed records are kept for each tube in every Armor and Artillery unit in the Army. Frequent replacement of gun tubes is undesirable not only for obvious fiscal reasons, but also due to the manpower and firepower drain it places on a unit in combat. One method of increasing the number of rounds a gun tube can safely fire is to place a metal lining inside the tube. One of the techniques for placing a lining inside a gun tube is called sputtering. Sputtering is a process where a cylindrical metal bar is inserted inside a gun tube, centered, anchored in place, and connected to a power source. The tube and rod are placed inside a chamber where all air is evacuated and replaced with a noble gas. When electricity is run through the source rod, a magnetic field develops and atoms from the metal source rod are broken away and literally fly across the gap from the source rod to the inside of the gun tube. During flight, the metal atoms collide with atoms of the noble gas that separate the source rod from the inside wall of the gun tube. Each collision with a noble gas atom causes the metal atom to change the direction of its flight and lose energy. In developing the sputtering process, the Army used three mathematical models to create computer simulations of the sputtering process. The computer simulations were used to quickly and inexpensively predict the effects of different materials and experimental conditions on the sputtering results. One model was used to simulate initial direction and energy of a sputtered atom. A second model was used to analyze the effects of an atom colliding with the inside of the gun tube wall. This chapter deals with the third mathematical model, which was developed to analyze the flight of a metal atom from the source rod to the inside of the gun tube. In order to model this portion of the problem, it is necessary to analyze the flight of an atom from the source rod (cathode) to the inside of the gun tube (anode) in
smallsteps.Eachsteprepresentsanimportanteventintheflight.Thefirsteventiswhentheatombreaksawayfromthecathode.Theinitialvelocityvectoroftheatomasaresultofthiseventisgivenasaninitialcondition.Thesecondeventisdeterminingthelocationofthesputteredatom'scollisionwithaneutralgas atom.Locating a collision site involvesdetermining afreepath lengthforthe atom and applying the free path length along the direction of the initialvelocityvector.Thethirdstepis examiningthe effectsof the collisionontheenergyandflightpathofthesputteredatom.Tosimplifythisstep,thecollisionisexaminedinacenterofmassframeofreference.Thefourthandfinal stepinthe process involves determining the location and energy of the metal atomwhen it impacts with the inside wall of the guntube (anode).Becauseasputtered atommight collidewithmultipleneutral gasatoms during its flightfromthesourcerodtothewalloftheguntube,stepstwoandthreemayberepeatedseveraltimesforeachatombeforecalculatingtheterminaleffectsinstepfour.GeometryWe begin with an explanation of the geometry of our model. The gun tube willbe treated as a hollow right circular cylinder with a radius equal to R. Thesource rod is also a right circular cylinder with a radius equal to r.Bothcylindersarecenteredonthezaxis sothatthestraight linedistancefromthesourcerod to the insideof theguntubeis equal to R-r.(SeeFigure 1)VsourcerodR-rguntubeyFigure188
88 small steps. Each step represents an important event in the flight. The first event is when the atom breaks away from the cathode. The initial velocity vector of the atom as a result of this event is given as an initial condition. The second event is determining the location of the sputtered atom’s collision with a neutral gas atom. Locating a collision site involves determining a free path length for the atom and applying the free path length along the direction of the initial velocity vector. The third step is examining the effects of the collision on the energy and flight path of the sputtered atom. To simplify this step, the collision is examined in a center of mass frame of reference. The fourth and final step in the process involves determining the location and energy of the metal atom when it impacts with the inside wall of the gun tube (anode). Because a sputtered atom might collide with multiple neutral gas atoms during its flight from the source rod to the wall of the gun tube, steps two and three may be repeated several times for each atom before calculating the terminal effects in step four. Geometry We begin with an explanation of the geometry of our model. The gun tube will be treated as a hollow right circular cylinder with a radius equal to R. The source rod is also a right circular cylinder with a radius equal to r. Both cylinders are centered on the z axis so that the straight line distance from the source rod to the inside of the gun tube is equal to R-r. (See Figure 1) Figure 1
Step1-lnitialVelocityVectorWewillusevectors inspherical coordinatestodescribetheflightofthesputtered metal atom. We use the angle to describe the rotationalangle inthex,y plane and the angle todescribe the elevation of theatom abovethex,yplane.Thus a unit direction vectorfor the sputteredatom willbe oftheform:Direction =(cos(0)·sin(Φ),sin(0)·sin(p), cos(p))工Direction of sputtered atomT1Figure 2Theunitdirectionvectoristhenmultipliedbytheinitialspeedtogenerateaninitialvelocityvectorfortheatom.Step2-LocatingCollisionSitesThelocationofanatom'scollisionisdeterminedbymovingaspecificdistancealongthedirectionofitsvelocityvector.Thedistancemovediscalledthefreepathlength.Thefreepathlengthchangesforeachatomaftereach collision.Thefreepathlengthisbasedonadistributionaboutanatom'smeanfreepathlength.Themeanfreepathlengthforan atomistheaveragedistancethat theatomwilltravelbeforeithasa collisionwithaneutral gasatom.Themeanfreepathlengthforallsputteredatomsofthesameelementremainsconstantaslongasthedensityoftheneutral gasremainsunchanged.The meanfreepathlength is calculatedbyexaminingtherelationbetweenthedensityof the neutral gas andthe distance a sputtered atom wouldtravel inaunit of timeif itwereallowed totravelunimpeded.The radiusofthesputteredatomandthedistanceittravelsinaunitoftimeareusedtocreateacylinderthroughwhichanatomtravelsduringtheunitoftime.Thevolumeofthecylinder is calculated and multiplied by the density of the neutral gas todetermine the average number of atoms that occupy a cylinder of that size. The89
89 Step 1 - Initial Velocity Vector We will use vectors in spherical coordinates to describe the flight of the sputtered metal atom. We use the angle q to describe the rotational angle in the x,y plane and the angle f to describe the elevation of the atom above the x,y plane. Thus a unit direction vector for the sputtered atom will be of the form: Direction = cos(q) ×sin(f), sin(q) ×sin(f), cos(f) Figure 2 The unit direction vector is then multiplied by the initial speed to generate an initial velocity vector for the atom. Step 2 - Locating Collision Sites The location of an atom’s collision is determined by moving a specific distance along the direction of its velocity vector. The distance moved is called the free path length. The free path length changes for each atom after each collision. The free path length is based on a distribution about an atom’s mean free path length. The mean free path length for an atom is the average distance that the atom will travel before it has a collision with a neutral gas atom. The mean free path length for all sputtered atoms of the same element remains constant as long as the density of the neutral gas remains unchanged. The mean free path length is calculated by examining the relation between the density of the neutral gas and the distance a sputtered atom would travel in a unit of time if it were allowed to travel unimpeded. The radius of the sputtered atom and the distance it travels in a unit of time are used to create a cylinder through which an atom travels during the unit of time. The volume of the cylinder is calculated and multiplied by the density of the neutral gas to determine the average number of atoms that occupy a cylinder of that size. The
lengthofthecylinderisthendividedbythenumberofatomsthatoccupythecylinderandtheresultisa reasonableapproximationforthelengthofthemeanfreepath (SeeFigure3).neutral gas atomssputteredmetalatomFigure 3DThis method yieldsthefollowing equation:L=WhereLis the mean元prfreepathlength,D is thedistancetraveled,pis the densityoftheneutral gasand ristheradius ofthesputtered metal atom.Oncethemeanfreepath isdetermined,thegraphinfigure4 is createdusing-ntodefinethecurve.2TheratiotheequationrepresentstheNNpercentage of free path lengths that are at least aslong as the correspondingnumberofmeanfreepath lengthsfromthegraph.Inthegraph,freepathlengthismeasuredintermsofthenumberofmeanfreepathlengthsasputteredatomwilltravelpriortoa collision (DIST/L),Dist ibutiondf FreePatts口TD3ADISTAFigure 4The freepathlengthfora specific atomat a specificmoment intime is thencreatedbyapplyingauniformdistributiontothen/Naxis (randomlychoosinganumberbetween0and1)andthenfindingthecorrespondingDiST/Lcoordinate90
90 length of the cylinder is then divided by the number of atoms that occupy the cylinder and the result is a reasonable approximation for the length of the mean free path (See Figure 3).1 Figure 3 This method yields the following equation: L D r = p r 2 . Where L is the mean free path length, D is the distance traveled, r is the density of the neutral gas, and r is the radius of the sputtered metal atom. Once the mean free path is determined, the graph in figure 4 is created using the equation n N e x L = - to define the curve. 2 The ratio n N represents the percentage of free path lengths that are at least as long as the corresponding number of mean free path lengths from the graph. In the graph, free path length is measured in terms of the number of mean free path lengths a sputtered atom will travel prior to a collision (DIST/L). Figure 4 The free path length for a specific atom at a specific moment in time is then created by applying a uniform distribution to the n/N axis (randomly choosing a number between 0 and 1) and then finding the corresponding DIST/L coordinate
onthegraphinfigure4.TheDIST/Ltermismultipliedbytheatom'smeanfreepath lengthtoyieldthedistancetotheatom'snext collision.Nowthatthedistancetothecollisionhas been determined,the actual location ofthe collision is found by multiplying distance to collision by a unit vector in thedirection of the sputtered atom's velocity vector and adding the resulting vectortothesputteredatom'soldpositionvector.Step3-EffectsofacollisionMovingtoa Centerof MassFrameof ReferenceWhena sputtered atom collides with aneutral gas atom,the sputtered atom willchange direction and speed. The analysis of a collisionis best made in a centerofmassframeofreference.Acenterofmassframeofreferencemeansthattheoriginofthecoordinatesystemused inmakingcalculations isfixedonthecenterofmass ofthetwoatoms involved inthe collision.Thus thecenterofmass doesnot move (as it would in the laboratory frame of reference). Using the center ofmass frame of reference makes the calculation of the changes in direction andspeed ofthe sputtered atomless complicatedbecausethechanges aremeasured againsta“stationary"point.ChangeindirectionSince we are working in spherical coordinates,the direction changeforcedupona sputteredatom as theresult of a collision witha neutral gas atomis bestexaminedintermsof anangleof rotationandanangleof deflection.Theangleofrotationisdeterminedbyexaminingthecollisionfromdirectlybehindthesputtered atom.Thekeytothe collisioneffects isthe relativepositions ofthecentersofmassofthesputteredatomandtheneutralgasatomatthemomentofimpact.lnfigure5,weseethatacircleofradiusequaltothesumoftheradiiofthe neutral gas atom and the sputtered metal atom (bmax) can be drawn centeredon the neutral gas atom.In orderfora collisionto occur,thecenterofmass ofthesputteredmetalatommustbelocatedwithinthecircleofradiusbmax.91
91 on the graph in figure 4. The DIST/L term is multiplied by the atom’s mean free path length to yield the distance to the atom’s next collision. Now that the distance to the collision has been determined, the actual location of the collision is found by multiplying distance to collision by a unit vector in the direction of the sputtered atom’s velocity vector and adding the resulting vector to the sputtered atom’s old position vector. Step 3 - Effects of a collision Moving to a Center of Mass Frame of Reference When a sputtered atom collides with a neutral gas atom, the sputtered atom will change direction and speed. The analysis of a collision is best made in a center of mass frame of reference. A center of mass frame of reference means that the origin of the coordinate system used in making calculations is fixed on the center of mass of the two atoms involved in the collision. Thus the center of mass does not move (as it would in the laboratory frame of reference). Using the center of mass frame of reference makes the calculation of the changes in direction and speed of the sputtered atom less complicated because the changes are measured against a “stationary” point. Change in direction Since we are working in spherical coordinates, the direction change forced upon a sputtered atom as the result of a collision with a neutral gas atom is best examined in terms of an angle of rotation and an angle of deflection. The angle of rotation is determined by examining the collision from directly behind the sputtered atom. The key to the collision effects is the relative positions of the centers of mass of the sputtered atom and the neutral gas atom at the moment of impact. In figure 5, we see that a circle of radius equal to the sum of the radii of the neutral gas atom and the sputtered metal atom (bmax) can be drawn centered on the neutral gas atom. In order for a collision to occur, the center of mass of the sputtered metal atom must be located within the circle of radius bmax