Electronic excitation in atomic collision cascades

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1 Nuclear Instruments and Methods in Physics Research B 8 (5) Electronic excitation in atomic collision cascades A. Duvenbeck a, Z. Sroubek b, A. Wucher a, * a Department of Physics, Institute of Experimental Physics, University of Duisburg-Essen, Universitaetsstrasse 5, D-5117 Essen, Germany b Czech Academy of Sciences URE, 1851 Prague 8, Chaberska 57, Czech Republic Abstract A model is presented which allows to incorporate low-energy electronic excitation into the molecular dynamics simulation of atomic collision cascades. The model treats the electronic energy loss experienced by all moving atoms as a source term in a diffusive description of the electronic excitation energy transport. In order to acknowledge (i) the large temperature gradients and (ii) the local lattice disorder induced by the projectile impact, the electronic heat diffusivity is allowed to vary as a function of space and time across the cascade volume. Ó Elsevier B.V. All rights reserved. PACS: 79..Ap; 71.1.Ca; 71.1.Li Keywords: Particle impact phenomena; Atomic collision cascade; MD simulation; Kinetic excitation 1. Introduction The impact of energetic particles onto a solid surface leads to the development of a collision cascade in the sub-surface region. It is well known that the atomic motion is accompanied by electronic excitation processes which manifest, for instance, in the observation of kinetic electron emission from an ion bombarded surface or the occurrence of excited or ionized states among the * Corresponding author. Tel.: ; fax: address: wucher@uni-essen.de (A. Wucher). flux of sputtered particles. In the past, several attempts have been made to incorporate excitation processes into the molecular dynamics (MD) computer simulation of atomic collision cascades. Since the ab initio treatment of a system large enough to enclose a kev-impact induced cascade is still prohibitively complex, simple concepts have been used which are mostly based on collisional excitation by electron promotion in close binary encounters [1 5]. We have recently developed an alternative scheme [1] where the friction-like electronic energy loss experienced by all moving particles is treated as a source term in a diffusion equation describing the spread of the generated X/$ - see front matter Ó Elsevier B.V. All rights reserved. doi:1.11/j.nimb..1.5

2 3 A. Duvenbeck et al. / Nucl. Instr. and Meth. in Phys. Res. B 8 (5) excitations [ 9]. In this approach, the electronic heat diffusivity D enters as a parameter which depends on the electron temperature, the lattice temperature as well as the crystallographic order within the solid. All three quantities are heavily modified in a collision cascade, leading to a pronounced space and time dependence of D. The numeric procedure used in [1] is based on a GreenÕs function method which permits a time dependence but does not allow for a spatial variation of the diffusivity. In this work, we extend the description to arbitrary variations of D by incorporating an explicit numerical integration of the diffusion equation.. Description of the calculation We use a standard molecular dynamics code described in detail earlier [11] to follow the particle kinetics within the collision cascade initiated by the impinging projectile. The Ag(1 1 1) model crystallite contains about 5 atoms distributed over 18 atomic layers. The interaction among the atoms is described by a MD/MC-CEM many-body potential [1]. The electronic system in our model is represented by a valence conduction electron gas, characterized by the Fermi energy E F and the electron mean free path k, into which the Ag crystallite is embedded. The electronic energy loss per unit traveled distance of the particle moving with the kinetic energy E k is described by the Lindhard electronic stopping power [13], leading to de k ¼ Kv ¼ AE k ; ð1þ where A is a constant evaluated as s 1 for an Ag atom moving in Ag metal. The amount de(r, t) of excitation energy that is transferred into the electronic system within is therefore given as deðr; tþ ¼ A X E i k ðtþdðr i rþ ¼AE k ðr; tþ; ðþ i where E i kðtþ denotes the kinetic energy of the ith particle moving at r i and time t. The spread of the excitation energy E(r,t) is described by a diffusion equation oeðr; tþ Dr Eðr; tþ ¼ ot deðr; tþ s ð3þ with a heat diffusivity D ¼ 1 3 kv F. As discussed in detail in [1], we disregard in (3) the energy transfer back from electrons to the lattice as a second order effect. The energy density resulting from the numerical solution of (3) is converted into an electron temperature using the electronic specific heat [1] c e ¼ p n e k b ¼ C ðþ T F (k b : Boltzmann constant, n e : electron density, T F : Fermi temperature). In contrast to the GreenÕs function method employed in [1], the numerical integration of (3) is performed explicitly using a finite differences approach by discretizing the simulation volume into cells of Dr =3Å dimension. The time step Dt is adjusted in order to ensure numerical stability according to the criterion Dt/Dr < D/. In some cases, this requires extremely small values of Dt down to about 1 s. Since under the conditions explored here it is not necessary to follow the atom kinetics with such time resolution, the MD simulation is run with its own dynamic time step (fs), and the atom positions and velocities are interpolated for the calculation of E. The boundary conditions for the solution of (3) are chosen as follows. At the surface, a Neumann condition is applied by setting the derivative of E along the surface normal to zero in order to prohibit diffusion into the vacuum. At all other boundaries of the simulation volume (Å) 3, the value of E is fixed to zero. At the beginning of the simulation, E is initially set to zero in the entire simulation volume. The central parameter of the model is the electron mean free path k determining the heat diffusivity D. Under non-equilibrium conditions characterized by an electron temperature and a lattice temperature T L, we use [15 17] v F k ¼ at e þ bt ; ð5þ L where the first and second terms in the denominator refer to electron electron and electron phonon

3 A. Duvenbeck et al. / Nucl. Instr. and Meth. in Phys. Res. B 8 (5) interaction, respectively. For silver, the Fermi velocity is v F = cm/s [1] and the constants a and b are estimated as a = K s 1 [1,18] and b = K 1 s 1 [1]. At room temperature ( = T L = 3K), the mean free path resulting from (5) is of the order of several 1nm. Under these conditions, a very high value of D 18cm /s results which would lead to a very rapid dissipation of all excitations into the bulk. In the course of the collision cascade, however, the lattice is rapidly heated to temperatures of the order of several thousand K, reducing k to several nm and, hence, the magnitude of D to about 1cm /s. Note that these values refer to a hot but otherwise unperturbed crystalline solid. As a second effect, the crystallographic order is severely modified in a collision cascade, leading to a transient state where the solid is completely amorphized in parts of the cascade volume. In order to estimate the heat diffusivity under these conditions, we assume for the fully amorphized state an electron mean free path of the order of one interatomic distance, leading to values of D 1cm /s. 3. Results and discussion The MD simulations were performed for a 5- kev Ag projectile normally incident onto an unreconstructed Ag(111) surface. In order to test the time consuming numerical solution of (3) and evaluate the role of different parameters, the calculation of electronic excitation is performed for one specific trajectory (impact point) which represents one of the largest events with a total of 8 sputtered atoms. In principle, the calculation described in Section results in a complete set of E or as a function of space and time in the solid. A 3-D visualization of such a distribution can be found at physik.uni-essen.de/wucher/. To visualize the results in a simple graph, we in the following plot the time dependence of the surface electron temperature as a function of the radial distance r from the impact point. A typical example of such a plot which is obtained for a constant diffusivity of D =cm /s is displayed in Fig. 1. In good agreement with our previous calculations [1] (1 3 K) D = cm /s r = Å Fig. 1. Surface electron temperature versus time after the projectile impact calculated for different radial distances r from the impact point. The calculation was performed using a constant heat diffusivity of D =cm /s. using the GreenÕs function method, the predicted excitation rapidly reaches a maximum induced by the passage of the projectile through the surface layer of wih Dr. During the time the projectile spends in this layer, a sharp initial rise of is followed by a small but extremely fast decay induced by the onset of diffusion. After the projectile has penetrated the surface layer, the original excitation which is mainly located at the impact point is rapidly spread both parallel to the surface and into the bulk. With increasing distance from the impact point, the initial peak is broadened and the temporal decay becomes less pronounced, a finding which reflects the increasing importance of slower heating by recoil atoms from the collision cascade. In the time interval between 1fs and 1ps where most of the sputtering occurs, the surface electron temperature assumes values of several hundred K which slowly decrease with increasing time. Note that the fine structure observed in the curves is not induced by numerical noise but reflects the statistics of particle motion in the collision cascade. Although valuable to prove the functionality of our numerical procedure, the assumption of a constant heat diffusivity is probably unrealistic. From

4 38 A. Duvenbeck et al. / Nucl. Instr. and Meth. in Phys. Res. B 8 (5) (1 3 K) D = D( (r,t),t L =7K) r = Å Fig.. Same as Fig. 1, the calculation now being performed with a heat diffusivity D( (r,t), T L = 7K) which varies according to the local electron temperature. (1 3 K) 1 8 D = D( (r,t),t L (r,t)) r = Å Fig. 3. Same as Fig. 1, the calculation now being performed with a heat diffusivity D( (r,t), T L (r,t)) which varies according to both the local electron and lattice temperature, respectively. the data presented in Fig. 1, it is evident that the electron temperature reaches values which are large enough to warrant a pronounced reduction of k and D. As a second step towards a more realistic description of the excitation profile, we therefore update the value of D at every time step and position by means of (5), using first only the actual values of (r,t) and keeping T L constant at 7K (corresponding to D =cm /s in the limit of low ). The results are displayed in Fig.. Due to the -induced diffusivity reduction, the initial excitation remains more localized and the maximum of is much more pronounced than before. At times larger than 1 fs, however, practically the same curves are found as observed in Fig. 1, reflecting the fact that in this time range the electron temperature is already too low to significantly alter the diffusivity. In a third step, we incorporate the influence of lattice heating by calculating a local lattice temperature from the average kinetic energy of atoms in each cell and also inserting the resulting T L (r,t) into (5). The electron temperature profile obtained under these conditions is shown in Fig. 3. Evidently, the lattice heating produces a strong localization of the excitation which manifests as a pronounced broadening of the initial peak. Note that the largest electron temperature observed close to the impact point amounts to almost 1 5 K! Moreover, the reduced diffusivity clearly produces a shift of the maximum towards larger time with increasing distance from the impact point. At 1 fs after the projectile impact, the surface electron temperature now assumes values of several thousand K which, however, still decay fairly rapidly to <1K at 75fs. As outlined above, the remaining influence to be taken into account is the lattice disorder introduced by the collision cascade. Due to the fact that the model allows for a space and time dependent variation of the diffusivity, it is in principle possible to implement a quantitative correlation between a local order parameter derived from the MD simulation and the diffusivity. The development of such an implementation is planned for the future but outside the scope of the present study. For a rough estimate of the effect, we therefore revert to the procedure already employed in [1] and assume D to vary linearly in time from to.5cm /s during the first 3fs and remain constant thereafter. The resulting surface electron temperature profile is depicted in Fig.. It is evident that the collision induced amorphization

5 A. Duvenbeck et al. / Nucl. Instr. and Meth. in Phys. Res. B 8 (5) (1 3 K) 5 3 D = -->.5 cm /s in 3fs r = Å r = 1Å r = 15Å r = 18Å show that the amorphization in the core of the cascade volume may have an even larger effect in localizing the excitation. Future work will therefore aim at incorporating the local and temporal disorder to calculate a realistic electron temperature profile, which will then be used to predict excitation and ionization probabilities of sputtered particles. Acknowledgements must lead to a strong localization of the electronic excitation within the cascade volume. In agreement with the results obtained in [1], the surface electron temperature is found to exhibit a second maximum at times where most of the sputtering occurs. The magnitude of in this regime of about 5K is large enough to induce electron emission and significantly influence the excitation and ionization state of sputtered particles.. Conclusion Fig.. Same as Fig. 1, the calculation now being performed with a homogeneous heat diffusivity D(t) which is assumed to vary linearly in time from to.5cm /s within the first 3fs and remain constant thereafter. Using a different numerical approach, the excitation model published earlier [1] has been extended to allow a temporal and spatial variation of the heat diffusivity. First results obtained for a selected MD trajectory demonstrate that particularly the lattice heating strongly affects the calculated electron temperature profile. Preliminary studies investigating the role of crystalline disorder The authors are greatly indebted to B.J. Garrison for providing the basis of the molecular dynamics simulation code. We also gladly acknowledge financial support from the Deutsche Forschungsgemeinschaft within the SFB 1. References [1] I. Wojciechowski, B.J. Garrison, Surf. Sci. 57 (3) 9. [] D.N. Bernardo, R. Bhatia, B.J. Garrison, Comp. Phys. Comm. 8 (199) 59. [3] R. Bhatia, B.J. Garrison, J. Chem. Phys. 1 (199) 837. [] Z. Sroubek, F. Sroubek, A. Wucher, J.A. Yarmoff, Phys. Rev. B 8 (3) [5] M.H. Shapiro, T.A. Tombrello, T.J. Fine, Nucl. Instr. and Meth. B 7 (1993) 385. [] G. Falcone, Z. Sroubek, Radiat. Eff. 19 (1989) 53. [7] G. Falcone, Z. Sroubek, Phys. Rev. B 39 (1989) [8] G. Falcone, Z. Sroubek, Phys. Rev. B 38 (1988) 989. [9] Z. Sroubek, Appl. Phys. Lett. 5 (198) 89. [1] A. Duvenbeck, F. Sroubek, Z. Sroubek, A. Wucher, Nucl. Instr. and Meth. B 5 (). [11] M. Lindenblatt, R. Heinrich, A. Wucher, B.J. Garrison, J. Chem. Phys. 115 (1) 83. [1] C.L. Kelchner, D.M. Halstead, L.S. Perkins, N.M. Wallace, A.E. DePristo, Surf. Sci. 31 (199) 5. [13] J. Lindhard, M. Scharff, Phys. Rev. 1 (191) 18. [1] C. Kittel, Introduction to Solid State Physics, John Wiley & Sons, 1971, p. 9. [15] C. Schaefer, H.M. Urbassek, L. Zhigilei, V, Phys. Rev. B () [1] X.Y. Wang, D.M. Riffe, Y.S. Lee, M.C. Downer, Phys. Rev. B 5 (199) 81. [17] D.S. Ivanov, L. Zhigilei, V, Phys. Rev. B 8 (3) [18] M. Kaveh, N. Wiser, Adv. Phys. 33 (198) 57.

The role of electronic friction of low-energy recoils in atomic collision cascades

The role of electronic friction of low-energy recoils in atomic collision cascades A. Duvenbeck 1 and O. Weingart 2 and V. Buss 2 and A. Wucher 1 1 Department of Physics, University of Duisburg-Essen,