APPLICATIONS OF THE MONTE CARLO CODE TRIPOS TO SURFACE AND .
Nuclear Instrumentsand MethodsNorth-Holland,Amsterdamin PhysicsResearchB28 (1987) 175-184175APPLICATIONSOF THE MONTE CARLO CODE TRIPOSTO SURFACE AND BULK ION TRANSPORT PROBLEMS *P.S. CHOUMechanical,Receivedand N.M.Aerospace6 JanuaryGHONIEMand Nuclear EngineeringDepartment1987 and in revised form 30 MarchUniversity of California, Los Angeles,California 90024, USA1987A general dynamic Monte Carlo ion transport code, TRIPOS (TRansportof Ions in POlyatomicSolids) is used in this work. TheTRIPOS code uses both power-lawcross sections and a newly developed solution to the scatteringintegral for treatmentoflarge-anglenuclear collisions. Small-anglenuclear collisions and electronic stopping are described as a continuousenergy loss inbetween large-angleion-atomcollisions. Applicationsof TRIPOS to surface, bulk, and deep penetrationproblems in multilayerpolyatomicmedia are given.1. Introduction and backgroundA sound understandingof the physical phenomenaassociated with ion transport in solids is critical to theadvancementof new and emerging technologies.Forexample, the first wall and divertor or limiter components of fusion energy devices experience high fluxes ofcharged particles and neutrons, which lead to a varietyof effects (i.e., sputtering, blistering, and bulk damage).The productionof primary knock-on atoms (PKAs) infission nuclear reactor materials leads eventually to thedegradationof design and safety-relatedmaterial properties. New material processing technologiesusing ionbeams or plasmas rely on a detailed knowledge of iontransport physics. Understandingof the physics of radiation damage in structural solids and microelectronicmaterials is critical to the developmentof space anddefense related technologies.Ion slowing-downprocesses can be attributed to ionenergy losses through different interactions (e.g., excitation and ionization of electrons, inelastic nuclear collisions, elastic nuclear interactions,and bremsstrahlungphoton emissions).There are two general classes ofapproachesfor the analysis of ion transport phenomena, the deterministicand probabilisticmethods. Deterministic approaches are based on either analytical ornumerical solutions to the ion transport equation. Onthe other hand, probabilisticsolutions rely on using theMonte Carlo method, where an ensemble of particles is* This work is supportedby the National Science Foundation,Grant#CPE-81-11581;the US Departmentof Energy,Office of Fusion Energy, Grant #DEFG-3-84ER52110withUCLA;and te State of Californiathroughthe MICROProject, Grant #UC-85-151with matching funds from TRWCorporation,Grant #TRW-A576778AB5S.0168-583X/87/ 03.500 Elsevier Science Publishers(North-HollandPhysics Publishing Division)B.V.used to simulate the ion interaction processes followingphysical laws.The ion transport equation is derived based on particle and energy conservationprinciples[l-6].Thetransport equation is quite complex, and analytical orstandarddiscretenumericalsolutionsfor generalgeometries and a large number of degrees of freedomare non-existent.In polyatomicsolids, coupled transport equations are required [5,6] which further increasethe complexity of analytical or numerical solutions ofthe problem.However, if a few ion parametersaredesired, simplified forms of the ion transport equationcan be used. Lindhard, Scharff, and Schiott [7] used themoments method by assuming continuous slowing-downprocesses to calculate the range distributionand associated parameters.Lindhard et al. [8] developed an integral equation relating radiation damage to depositedenergy and atomic displacement.Parking and Coulter[9] extended the treatment for the atomic displacementcalculation in polyatomic solids. However, these calculations neglect the spatial distributionof radiationdamage.Sanders [lo] derived a vector range distribution function. For one-dimensionalgeometry, Sigmund [ll] expanded the vector range distributionin Legendre polynomials in moment solutions. Brice [12] developed anintegral equation describing the energy deposition profiles using a similar approach. All of the above methods,however, use simplified forms of the ion transport equation where detailed particle phase-spaceare not included. Hoffman et al. [5] used the discrete-ordinatemulti-energy-groupmethod, which was originally developed for neutron transport [13,14], for the analysis ofsurface sputteringproblems.Using approximatecoupled diffusion equations, Chou and Ghoniem [6] devel-
116P.S. Chou, N.M. Ghoniem /Applications of TRIPOS to surface and bulk ion transportoped an analytical method for the slowing down ofion-induced cascades in precipitate dissolution problems. The diffusion equations were de-coupled using theNeumann series expansion technique.To describe the many-body interaction of N atomsin the crystal, 3N Newtonian equations of motion arerequired for the description of the crystal. A PKA eventcan be initiated by giving an atom a kinetic energy Ewith the momentum gained in a specified direction.Numerical procedure is then used to integrate theseequations of motion until equilibrium is reached. Thesecalculations are fully dynamic since all degrees of freedom are allowed to vary simultaneously to satisfy therequirements of interaction forces and the Newtonianlaws of dynamics. This many-body integration methodis very useful in demonstrating directional effects suchas focusing and channeling. The major limitations liewith the maximum available memory capacity and computation speed of a computer. Consequently, only asmall number of low energy PKAs can be simulated inthis type of analysis [15]. This method is generallyreferredto as moleculardynamics(MD);theMARLOWE code [16,17] is an example of an MD code.At PKA energies above few hundred eV, a simplifiedapproximation can be made. Instead of the many-bodyinteraction problem, the ion-atom interaction is treatedas a binary collision approximation (BCA). The MonteCarlo method is used within the framework of the BCA.The Monte Carlo method can accurately simulate particle behavior in solids, with modest increments in computational difficulties for each added degree of freedom.This method is particularly suitable for applications of“supercomputers.” In the Monte Carlo method, sampling is conducted from probability distribution functions according to relevant physical laws.Beeler [18,19] applied the assumption that the atomiccollision cascade can be described as a branching sequence of binary collision events to ion transport calculations. Yoshida [20], Oen and Robinson [16,17], Ishitaniet al. [21], Robinson and Agamy [22], Biersack andHaggmark [23], Ziegler et al. [24]. Attaya [25], and Chouand Ghoniem [26] developed Monte Carlo codes basedon the binary collision approximation. Roush et al. [27]as well as Moeller and Eckstein [28] used the sameconcept to develop time-dependent Monte Carlo codes.In this work, a dynamic binary collision Monte Carloion transport code, TRIPOS, is presented. It is a generalpurpose code for the efficient calculation of ion transport and effects in a multi-layered structure composedof polyatomic solids. TRIPOS can be applied to theanalysis of bulk or surface time- and fluence-dependentproblems. Thus, complex surface evolution problemscan be analyzed. Importance sampling techniques areincorporated in order to enhance the computation speedand reduce the variance in the results.The validity of the results from ion transport simula-tions depends heavily on the ability to accurately describe both the interatomic collisional behavior and theelectronic energy loss. In order to correctly predict theion collisional behavior in solids, the interatomic potential has to be known with great accuracy. However, theoverlapping and shielding effects of the electron cloudsduring the atomic collision process make a completedelineation of the interatomic potential difficult. Manyinteratomic potential models were proposed in the literature. Models most widely used include the ThomasFermi [29,30], Bohr [31], Born-Mayer [32], Moliere [33],and Ziegler [24] potentials. However, these complexpotentials result in difficulties for analytical solutions ofthe scattering integral. Biersack et al. [23,24] used afitting function to the numerical results from thescattering integral to approximate atomic scattering. Inthe present work, two methods of calculating scatteringproperties are used. The first is based in approximatepower-law fits to the Thomas-Fermi potential. In thesecond method developed by Blanchard, Ghoniem, andChou (BGC) [34], a second order Taylor expansionabout the distance of closest approach is used for theintegrand of the scattering integral resulting in accurateanalytical solutions.2. Theoretical background for the TRfPOS codeIn this section, we present a brief review of thetheoretical background for the TRIPOS code. The interested reader should consult ref. [26] for more detail.Several new theoretical features have been implementedin the code since the original work (261 and their features are discussed in this section.2. I. Brief description of previous theoretical featuresLindhard et al. [7] used the momentum approximation to solve for the scattering integral based on thepower-law potentials. The solutions from the powerpotentials are applied to the derivation of the so-calledpower-law cross sections. This technique bypasses theslow and costly process of numerically solving thescattering integral. In the power-law approximation, thedifferential cross section is given by [3S]da( E, T) C,,,E-“T-I-“’dT,(1)where C,,, is a constant and dependent on the ion-targetcombination and the energy regime. The total nuclearscattering cross section is expressed in the form:o(E) ‘[T;“-(AI?-“], C,, ln(AE/T,),m O,(2)m O,(3)where A is the maximum fractional energy transferredin a collision, and T, is the lowest energy transferred. At
P.S. Chou, NMGhoniem /Applicationshigh energies this characteristic energy, T is taken tobe either the surface binding energy or the displacementthreshold energy, depending on the application. However, for cases when the total cross section is so largethat the mean-free path is less than a lattice constant, amean-free path of one lattice constant is assumed. Thecross section corresponding to a mean-free path of onelattice constant is a0 ?rr,‘, where r,, is one half of thelattice constant. Eqs. (2) and (3) are used to solve for T,using ee. This procedure is also used for the cases wherethe mean-free path is excessively large to alleviate theerroneous accounting of electronic and small energynuclear losses.The free path between nuclear collisions can besampled from an exponential distribution based on thetotal cross section. However, if the free path is smallerthan r,, no sampling is used and the mean-free path isemployed instead. This conserves the free path distribution, on the average, and does not violate the BCA. Thecollision probability of an incident ion with one type ofpolyatomic specie is proportional to both the atomicdensity and the microscopic scattering cross section ofthat specie. The scattering angles of the incident ionand a recoil are related to the energy transferred in acollision. The scattering of ions from background atomsis significant. For collisions with background electrons,however, the momentum exchange is so small that thetrajectories for ions are not affected. The energy loss ofan incident ion resulting from the interaction with electrons can be adequately treated using an electronicstopping power equation that combines the LindhardScharff formula at low energy and Bethe-Bloch equations at high energy [23,24]. In the intermediate energyrange, the equation is consistent with experimental observations.To conserve the total nuclear energy loss, the contribution of small-angle nuclear collisions (i.e., energytransfer less than T,) has to be considered. A small-anglenuclear stopping cross section is used to take account ofthe low energy nuclear collisions. It has the form:S”,(E)S,,(E) &E-“(zy2In 2,( c)- T,-“),in 1,(4)m l,where T, is a cutoff energy of the order of a few eVs.For the case when T, is less than T,, the small-anglenuclear stopping cross section is taken to be zero.Therefore, the total “continuous” energy loss, AE, hascontributions from both electronic stopping as well assmall-angle nuclear collision. The energy loss, A E, takesthe formAE AICN,[S,(E) S,,(E)],(6)of TRIPOSto surface and bulk ion transport117where S, is the electronic stopping cross section, N, theatomic density for the i th specie, and AI is the freepath between “large-angle” nuclear collisions.2.2. AtomicscatteringTheoretical treatments of atomic scattering in theBCA start with the scattering integral which has theformP drB a-2jmPr2111V(r)---4Iy221where 6’ is the deflection angle in the center of masssystem, p the impact parameter and p the distance ofclosest approach, E, the center of mass kinetic energy,and V(r) the interatomic potential. Lindhard et al. [7]used the momentum approximation to simplify thescattering integral. Their approximation for the newscattering integral takes the form0s-EkDV[(x2 p2)“2]dx,(8)where E is the ion energy in the laboratory system. Forpower-law potentials with s 1 (pure Coulombic) ands 2, the exact scattering integral can be analyticallysolved. The results of eq. (8) agree well with those fromthe exact scattering integral [eq. (7)] for small scatteringangles. However, for near head-on collisions, themomentum approximation yields larger scattering angles.The power-law representations of interatomic potentials are derived from appropriate fits to the ThomasFermi potential. Moliere [33] and Ziegler et al. [24] haveattempted different forms of exponential screeningfunctions to fit interatomic potentials. However, thescattering integral using these potential functions cannot be analytically solved.The contribution to the deflection angle of the incident ion is considerable when the interacting particlesare separated by distances in the vicinity of the point ofclosest approach; whereas the contribution is negligiblewhen the two particles are separated by larger distances.Based on this concept, Blanchard, Ghoniem, and Chou(BGC) [34] expanded the interaction potential in eq. (7)about the point of closest approach to the second orderof a Taylor series expansion. The potential is fullytruncated beyond a separation distance of more than afew impact parameters. Analytical solutions of the exactintegral [eq. (7)] were obtained for both the Moliere andZiegler potentials.Fig. 1 shows a comparison between the scatteringangle results from the power-law approximation, theBGC analytical solution [34], and exact numerical calculations using Ziegler’s universal screening function withdimensionless energies, 6, spanning a range of over five
P.S. Chou, N.M. Ghoniem / Applications of TRIPOS to surface and bulk ion transport178potential barrier model is used for slab geometry. Thebinding energy U at an ejection direction cosine withplane normal p as in the planar model is given by [27]:fJ(p) Q/r23(9)where U, is the minimum energy for particle historytermination.For spherical geometry, the isotropic model[11,27] is used,U(IL) u,.‘-0.040a.016.012.0200IMPACT PARAMETER0 Power Law0 Zegler’s(Nun-dA Ziegler’sCBGCIFig. 1. The scattering angle as a function of impact parameterusing power-law and Ziegler’s universal potentials at dimensionless energy [eq. (S)] of 10m5, 3 x 10m4, 3 X 10-3, lo-’ and3. The scattering integrals are numerically integrated for bothpower-law and Ziegler’s universal potentials. The approximateintegral method by BGC [34] is also used to solve for Ziegler’suniversal potential.orders of magnitude. Calculations were also made usingthe Moliere potential [34]. The results show the following.(1) The “exact” form of the screening function has themost significant effect on the solution of scatteringintegral. This is clear when we compare Molierepotential results to calculationswith Ziegler’s universal potential.(2) The power law is a reasonable approximation to thescattering process. Its major advantage lies in itssimplicity within the context of the Monte Carlomethod.TheBGC analytical solution is accurate and within(3)a few percent of the numerical results with impactparameter up to 25 screening lengths for both theMoliere and Ziegler potentials.Because of the relative simplicity of the power-lawpotential results and the fact that collisions are nearCoulombic at high energies, the majority of the calculations in TRIPOS are performed in this mode. However,for verification,an option using the BGC method isprovided.2.3. Particle history terminationand sputteringA particle history is terminated in two cases: (1) if itsenergy falls below a minimum value, which is the displacementthresholdenergy (Ed) in the bulk, or thesurface binding energy near the surface; or (2) if itphysically leaves the region of interest.Calculations of sputtering into vacuum are sensitiveto the surface binding energy. In this work, the planar(10)To account for all sputtered particles, ions with energygreater than U, are simulated. However, it is knownthat recoils with energy less than the displacementthreshold energy are not permanentlydisplaced fromtheir original lattice positions. Cascade simulations byHeinish [36] using the MARLOWEcode [16,17] indicated that a displaced atom has to be at least a distanceof 6.5 lattice constants away from the vacancy site to bepermanentlydisplaced.Based on this conclusion,wedefine an equivalentsurface binding energy, U,,, asfollows:Uo, UO x(Ed-U,)/X E,,for x X,for x A,(II)where E,, is the displacementenergy, x is the depth ofrecoil generation,and X is a distance of 6.5 latticeconstants.The use of this equivalent surface bindingenergy results in terminationof all bulk recoils withenergy less than E,, . This can greatly reduce the numberof simulated recoils without affecting the displacementdamage or sputtering results.2.4. Dynamic surface evolutionThe evolution of an alloy surface is both time andflux dependent because of the ion beam induced changesin surface compositions.In the TRIPOS code, thesurface regions are divided into many layers, the thickness of each being a small fraction of the incident ionprojectedrange. For each layer, atomic species conservation is required to trace time and fluence dependencies. During the simulation, pseudo particle historiesare used with each history representinga fluence of IV,which is on the order of 1012 cmm2. Each layer isgenerally represented by N pseudo-particles.Conservation requires thatN nAt/W(12)for each layer, where n is the atomic density and At isthe thickness of the layer. Particle balance is performedfor each layer in the events of sputtering, dissolution,and implantation.Such balance yields informationonthe local layer composition.Also, other phenomenaindynamic surface evolution processes require additionalmodeling.For example, the recoil implantationandmixing can cause the formationof local superdensematerial which eventually leads to local relaxation and
P.S. Chou, N.M. Ghoniem / Applications of TRIPOS to surface and bulk ion transport119expansion. This process is related to the phenomenonofrecoil mixing. The superdensesolid is modeled to expand homogeneouslyuntil the theoreticaldensity isreached.hand, for r Z,,/Z,, the particle is eliminated and discarded from further consideration.Statistically,thistechnique also conserves the importanceof the entiresystem [37-401.2.5. Variance reduction and importance sampling3. ResultsThe standard deviation, which is the square root ofthe variance, is used as a measure of the statisticaluncertaintyin the Monte Carlo results. The variancedecreases with the number of simulated histories. Theuncertaintyin simulated average results decreases withthe square root of the number of particle histories used.Since the cost of computationcan be approximatelyregarded as a linear function of the number of histories
Lindhard et al. [7] used the momentum approxima- tion to solve for the scattering integral based on power-law potentials. The solutions from the power potentials are applied to the derivation of so-called power-law cross sections. This technique bypasses the slow and costly process of numerically solving the
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