Analytical Thermal Spike (ATS)

1. Ion H H H H (*) He He He He (*) Energy [MeV/n] 0.5 0.3 0.2 0.11 6.5 3.25 1.8 0.6 dE/dx [eV/Å] 3.78 5.23 6.5 7.47 7.42 11.2 15.5 21.7 E0 R0 R0 y 158.5 219 272. 5 313 311 470 650 910  t 0 N Analytical theories • Great number of analytical models • Large number of approximations • Based on thermodynamics considerations e.g.: - Gas Flow - Shock wave or pressure pulse - Thermal spike • n depends on the models(1, 3/2, 2, 3) Isotropic random kick • Principle : The energy is transferred to atoms (or eventually extracted from atoms) by a random isotropic kick processes: A kick of energy E0 instantaneously changes the atom momentum according to the equations: • (1) • - vi and vf are respectively the velocity before and after the kick, • - M is the atom mass, • E0 is the energy transferred to the atom and may be negative, • - q is an isotropic momentum. It is randomly tossed up in respect for the set of equations (1). q vi vf Before Kick After  = 10 ps t = 20 ps  = 0 s t = 20 ps He (0.6 MeV/n) t = 30 ps H (0.11 MeV) t = 30 ps Dn Abstract Due to its complexity, modelling of damage induced in solids by swift heavy ions was first undertaken using analytical theories based on model concepts such as thermal spike or pressure pulse model. About 10 years ago, Molecular Dynamics (MD) simulations were started in this field [1], reducing the number of approximations in the theoretical description. However, the electronic processes, which govern the primary ion-solid interaction and also the subsequent energy transfer from the target electrons to the lattice, are introduced in these simulations in an ad hoc manner. For instance, sputtering predictions are generally obtained considering that a fixed fraction of the projectile energy loss is deposited inside a cylinder of fixed radius. Thus, effects related to the radial extension of the electron cascade (e.g. velocity effect) can not be taken into account. Furthermore, for high-energy depositions, the sputtering yield is predicted to be proportional to the stopping power which is in disagreement with experimental data. The aim of our MD calculations is to explore the influence of the time and space dependent energy transfer from the electrons to the lattice atoms on observable quantities (e.g., sputtering and track size). For this purpose, we use the electron-phonon coupling mechanism developed by Toulemonde et al. [2] in the thermal spike model. The evolution of the electronic temperature Te(r,t) is described by a continuum equation that is coupled to the Newton equations of the target atoms (MD). We can assess, for example, the influence of the projectile velocity and of the electron phonon coupling on the atom dynamics. 1 H. M. Urbassek, H. Kafemann, R. E. Johnson, Phys. Rev. B 49, 786(1994) 2 M.Toulemonde, J. M. Costantini, Ch. Dufour, A. Meftah, E. Paumier and F.Studer Nucl. Instr. Meth. B116 (1996)37 Molecular Dynamics (MD) MD and ATS comparison Analytical Thermal Spike (ATS) e.g.: P. Sigmund & C.Claussen (81); R.E. Johnson & R Evatt (80) Principle  Solving the Newton equation for N atoms Principle • Local atomic equilibrium + Thermodynamics principle • Cylindrical symmetry T = T (r,t) • Diffusive transport of energy C(T) specific heat ; K(T) thermal conductivity • Initial temperature profile = Gaussian • Fixed width • Total energy = Stopping power • Sputtering = sublimation process Experiments Applied within the same conditions - Condensed-gas solid : Argon - Same initial temperature profile: (Cylinder or Gaussian profile) e.g.: n=1 for condensed Ar n=2 for condensed O2 n=4 for LiF  n function of the target Molecular dynamics  Analytical Thermal Spike • Advantages (/Monte Carlo Simulation) • Collective effects (Shock wave) • Adapted to strong perturbations • Anisotropic effects (focusson) • Disadvantages • Computing time is huge!!! • Small samples (N104 to 105 ) • Simple system model (Lennard-Jones) [H.M. Urbassek et al. 94] Analytical Thermal Spike = Incorrect Explanations • Velocity distribution Non Maxwellian • Transport of energy Not Only diffusive • Effect of pressure wake • Energy transport to the surface… Tsurf>Tbulk Result [E.M. Bringa et R.E. Johnson 98-99] Molecular Dynamics of damage and sputtering induced by swift heavy ions M. Beuve$&; N. Stolterfoht&; M. Toulemonde§; C. Trautmann$ ; H.M. Urbassek# § CIRIL (Centre Interdisciplinaire de Recherche Ion Laser), Caen, France; $ GSI (Gesellschaft für Schwerionenforschung), Darmstadt, Germany; & HMI (Hahn-Meitner-Institut), Berlin, Germany; # University of Kaiserslautern, Germany Why does MD give n =1 ? Time-dependent energy transfer Principle • Energy is progressively and periodically introduced during a time  with a period  /N (N from 100 to 1000).  Each transfer of energy is equal to(Lz is the thickness of the target) Effect of potential? Lennard-Jones n=1 Other potentials n1 Morse potential[E.M. Bringa et al. 00] • 1 parameter more to change stiffness • Obtained n~1.5 but: • Potential ~ hard sphere • Shockwave • Artificial ? • For each energy transfer i, only the atoms that set in the solid and within the cylinder of axe, the ion trajectory, and of fixed radius R0 (R0 =2) received an energy E0(i). Then (Nat(i) is the number of atoms in the solid setting in the cylinder) Effect of energy deposition ? Experimental argument =>Velocity effect Theoretical argument =>Work of Toulemonde et al. ....In this work , we investigated someeffects of the energy deposition • The energy transfer to the atoms is performed according to the previously described random kick process • Observations / Interpretations • For time scale lower than 10-12 s, the effect of a time-dependent energy transfer is irrelevant • For a time scale of 10-11s : • the power exponent is clearly modified. • for y 1 energy diffuses and can no longer produce an efficient sputtering • for y >> 1 less energy is taken away by non linear mechanisms (pressure pulse...). The sputtering is then more efficient Scaling to other materials Lennard Jones potential  Time scaling e.g.: Cu: tCu= 0.17 tAr • Cu atom dynamics is faster • Time effect should appear sooner Such time scales match, for instance, with F-centre creation in alkali halides.[N. Itoh et K. Tanimura 90] Thermal spike model of Toulemonde et al. Effect of energy deposition … F. Seitz et J.S.Koehler (23) ; M. Toulemonde et al. (92) The Idea:  To check the effects of a more realistic energy deposition beyond the fixed radial profile, we consider the thermal spike developed by Toulemonde et al. • Principle :  Based on two coupled equations: • one for the atomic subsystem • - Similar to the ATS model • onefor the electronic subsystem consisting of: • - A(r,t), takes into account : • i) Electronic excitations by the incident ion (10-18 s) • ii) Transport of the excitations by electronic cascades (10-18 ~ 10-15 s) • - A heat equation dealing with thermal electrons • g(Te-Tat) represents energy transfers due to electron-phonon coupling (g is constant) • Parameter of the simulation: • - Atomic subsystem : Solid argon • Potential : Lennard-Jones • Electron subsystem : Insulator • C(T) = 1 J cm-3 K-1 • K(T) = 2 J cm-1 s-1 K-1 • Electron phonon-coupling : Very strong • g = 0 for Tat > Te • g = 2 10 14 W cm-3 K-1 for Tat < Te • Close to the value for SiO2[Toulemonde et al. 2000] • Observations / Interpretations • Effect of a rather realistic energy deposition: • a velocity effect can clearly be observed • the power exponent is hugely modified • the area concerned by sputtering is quite large. Its radius reaches a value up to 11 (larger than the Bohr adiabatic radius) • although the atomic temperature stays lower than the sublimation temperature (Tsub= 640 K = kB-1.Us=55 meV.kB-1), the yield can reach very large value (e.g.: 150 at/ion) • for high dE/dx, the sputtering occurs in a collective way. A block of matter suddenly flows out (in less than 20 ps). • A very large value of electron-phonon coupling • Energy transfer occurs before: • any thermal diffusion of the electronic energy • any local thermodynamic equilibrium for the atomic subsystem. The notion of specific heat could not be used here to describe the atomic evolution • [E.M. Bringa et R.E. Johnson 98] • A shock wake takes away energy from the track • Other observations / Suggestions • The analysis of some movies seems to indicate that: • for H(0.11MeV), a significant part of the sputtered atoms would be ejected perpendicularly to the surface • for He(0.6MeV/n), more atoms seems to be ejected at large angles, • Indeed, in the latter case, a block of matter, which contains a great number of atoms, undergoes an evolution in the vacuum soon after its ejection: • - a kind of explosion occurs because the atoms, surrounding it in the solid, are now missing • - the highly perturbed surface attracts the last ejected atoms and then deviates them to higher angles • Both these phenomena would produce a more isotropic distribution. • These suggestions, which are qualitatively in agreement with measurements performed for LiF target (except for the sharp peak at 0°), have to be confirmed by angular distribution calculations. Some details: - The sample is decomposed in n cylinders Ci of axe, the ion trajectory and of radius Ri = iR0 , i [0, n-1] - n-1 domains Di are defined as Di = Ci\ Ci-1. A last one is defined as Dn = sample \ Cn-1 - At each time step t of the whole simulation (electron and atomic dynamics) and for each domain, - the atomic temperature Tat is evaluated - a transfer of energy is performed between both subsystems according to the formulae (Vi volume of Di ) The energy is given to (or taken from) the atoms according to the described random kick process. Note: outside the sample, g is set equal to 0 to solve the electron dynamics even far away from the ion trajectory Advantages ( / Analytical Thermal spike): - i) Includes electron dynamics instead of a simple fixed profile of deposited energy. - ii)Propose a process of energy transfer from electrons to atoms: electron-phonon coupling - iii) Takes account for phase transformation (melting, superheating…) y • Limits: • Assumes local atomic equilibrium • Does not take account for any mechanism such as shock wave, focus on.... • Considers the target as an infinite medium • Assumes sputtering as an evaporation process Bragg Peak (*) • General conclusion • From this work, we learned that it may not be realistic to model swift-ion interaction with solids depositing suddenly a fixed part of the stopping power in a cylinder of fixed radius. • We showed that a more realistic time-dependent and space-dependent energy deposition may drastically modify the sputtering yield. • In particular, for the first time, non-linear relations between yield and stopping power were showedwith Molecular Dynamicfor high stopping power. • This work opens directions to new extensive studies. Indeed, the whole process of swift-ion interaction with solid (including electronic excitations, transport, eventual trapping…), has, a priori, to be considered.. ? ?