Fragmentation simulations of C 60

1. Fragmentation simulations of C60 Titus A. Beu Co-workers: Horváth Lóránd, Ioan Ghişoiu, University ”Babeş-Bolyai” Department of Theoretical and Computational Physics Cluj-Napoca, Romania

2. Interest for fullerenes • 1985– H. Kroto, J. Heath, S. O'Brien, R. Curl, and R. Smalleydiscovery of C60- study of the absorption spectra of interstellar dust. • 1990 – W. Krätschmer, L. Lamb, K. Fostiropoulos, and D. Huffman production of pure C60 in large quantities. • 1996– R. Curl, H. Kroto, and R. SmalleyNobel Prize in Chemistry for the discovery of fullerenes. • Fullerite - new form of solid carbon - weakly bound C60 molecules (besides diamond and graphite). • Superconductors (Tc ~20-40 K) - alkali-doped fullerides with K or Rb. • Building blocks for nanoelectronics • Polymers obtained by electron-beam irradiation show metallic behaviour.

3. Interest for fullerenes Point group: Ih (icosahedron) Symmetry elements: 1 inversion point 6 equivalent C5 axes through opposite pentagons 10 equivalent C3 axes through opposite hexagons 15 equivalent C2 axes through opposite bonds 12 pentagonal faces – opposed pairs 20 hexagonal faces – opposed pairs 60 pentagonal edges surrounding pentagonal faces 30 hexagonal edges between hexagons.

4. Interest for fullerenes • Previous interest: electronic structure, IR and Raman spectra and metallic behaviour of electron-irradiated fullerene polymers • Present interest: understanding laser induced fragmentation of C60 • Fragmentation – complex electron transfer processes, ionization most important • Can be regarded as inverse to fullerene formation • Ubiquitous presence of fragmentation in nature

5. Previous publications • L. Horvath, T.A. Beu,”Tight-binding molecular dynamics simulations of radiation-induced fragmentation of C60”,Phys. Rev. B 77, 075102 (2008). • T.A. Beu, J. Onoe, ”First-principles calculations of the vibrational spectra of one-dimensional C60 polymers”,Phys. Rev. B 74, 195426 1-6 (2006). • T.A. Beu, J. Onoe, A. Hida,”First-principles calculations of the electronic structure of one-dimensional C60 polymers”,Phys. Rev. B 72, 155416 (2005). • T.A. Beu, J. Onoe, K. Takeuchi, “Structural and vibrational properties of C36 and its oligomers (C36)M=2,3,4 by tight-binding molecular dynamics”,Eur. Phys. J. D 17, 205-212 (2001). • T.A. Beu, J. Onoe, K. Takeuchi, “Simulation of Raman spectra of C60 and C70 by non-orthogonal tight-binding molecular dynamics”,Eur. Phys. J. D 10, 391-398 (2000).

8. Fragmentation and ionization of C60by femtosecond laser ablation 2.7 x 1011W/cm2 6.2 x 1011W/cm2 2.4 x 1012W/cm2 Kobayashi et al., J. Chem. Phys. 126, 61101 (2007). • Reflectron TOF spectrometers – only charged species can be detected • Laser pulse: 796 nm, 200 fs • Laser spot diameter (20 µm) >> C60 diameter • Bimodal fragment distribution: Cn and C60-2n • Multiple ionization (up to C605+) – multiphoton ionization – highly excited fullerene

9. Nonorthogonal tight-binding MD One-electron energies k: H and S - representation of some nonorthogonal set of atom-centered orbitals. Papaconstantopoulos et al.– Phys. Rev. B 62, 4477 (2000).Etot is the sum of (shifted) one-electron energies: Interatomic forces:

10. Nonorthogonal tight-binding MD Bond orbitals are constructed from sp3 hybrids Fullerenes – the nominal sp2 bonding occurs on a curved surface → sp3 bonding For a molecule composed of N atoms – n = 4Nelectrons, nocc = n/2 and nk = 2. S has a similar form, with diagonal elements equal to 1.

11. Nonorthogonal tight-binding MDTB parametrization of Papaconstantopoulos et al., Phys. Rev. B 62, 4477 (2000). Pseudo-atomic density – environment of each atom: Cutoff function: On-site terms (diagonal terms of H) – Birch-like equation (l = s,p): Slater-Koster hopping parameters ( = ,) Forces can be expressed analitically → tractable large-scale MD simulations.

12. C2hsymmetry C60 polymers For C60D is the cage extent, d1 and d2 are the minimum and maximum bond length, respectively. For the C60 polymer, D is the length of the translation vector. For C240 the central values are specified.

13. Simulation details Improved model • Ionized fragments • Non-adiabatic excitation of C60 • Measurements in terms of two parameters: • excitation energy – exp. deposited energy – up to 1000 eV (1.66 Ebind) • total charge – up to +24  ionization state of fragments • Ensemble averages over 200 trajectories for each parameter combination • Time steps = 0.05 – 0.5 fs Initial C60 geometry:simulated annealing • Bond lengths: 1.41 Å and 1.44 Å (exp. 1.37 Å and 1.42 Å) • HOMO-LUMO gap: 1.71 eV (exp. 1.6 – 1.85 eV)

14. Simulation details Excitation: • Experiments – laser pulses from cca. 5 to hundreds of fs • Ramp-like input of excitation energy and charge – texc = 0, 0.1, 0.5 ps • At each time step – equal elementary excitations random corrections of atomic velocities • At equal number of steps – integer charge is added (<= total charge) • If number of ramp steps < total charge – several elementary charges added at each ramp step • If fullerene fragments during ramp – charge added to random fragment

15. Simulation details Charge (re)distribution: Basic model: integer charge on fragments • Charge on atoms can be fractional • Charge of new fragments – sum of partial atomic charges in originating fragments • Fragments are ordered increasingly according to deficit to integer charge • Fragments with minimal deficit rounded up, rest rounded down • Total ionization state remains unchanged after excitation Simple alternative model:fractional charge on fragments • Constant equal fractional charge on atoms • Can be considered time-averaged charges

16. Simulation details Charge (re)distribution: 2/4 + 2/4 + 2/4 + 2/4 2/3 + 2/3 + 2/3 1/2 + 1/2 Charge = 2 + 2/12 Deficit = 10/12 Final charge = 2 Charge = 1 + 1/3 Deficit = 2/3 Final charge = 1 Charge = 1 + 1/2 Deficit = 1/2 Final charge = 2

17. Simulation details Fragmentation criteria: • Identification of fragments: recursive labeling algorithm • Fragmentation criteria are checked every 1000 time steps, first after 3000 steps (~1.5 ps) • First stopping criterion fulfilled if labels of atoms do not change between consecutive fragmentation checkpoints • Second stopping criterion fulfilled if distance between any fragments is larger than extent of tight-binding C-C potential well (~4Å) – recombination probability is negligible

48. Fragment size distributionsNeutral fragments • Ensemble-averaged distributions • Fragmentation sets in at ~ 80 eV • Gradual shift of the distribution maximum to small clusters • U-shapedistribution for medium energies – C1 absent • Bimodal size distributions – caused by C2 evaporation • Most probable clusters: C2 & C3 • Power-law dependence for high energies (> 1000 eV) • Full dissociation above 5000 eV • Profiles similar to fragmentation experiments with projectiles

49. qtot= 10 Fragment size distributionsCharged fragments +10e +20e qtot= 20 No U-shape distributions – strong Coulomb repulsion within large fragments leads to successive disintegration into smaller ones.

50. Average fragmentation probability 100% fragmentation 0% fragmentation • Phase transition(50% fragmentation) at ~90 eV for C60 and ~0 eV for C60+24 • Transition region becomes narrower with increasing total charge • Increased texc shifts transition to higher energies