Atomistic simulation and comparison with timeresolved

Transcription

1 Vibrational properties of metal nanoparticles: Atomistic simulation and comparison with timeresolved investigation Huziel E. Sauceda 1, Denis Mongin 2, Paolo Maioli 2, Aurélien Crut 2, Michel Pellarin 2, Natalia Del Fatti 2, Fabrice Vallée 2, Ignacio L. Garzón 1* 1 Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal , México, D. F., México 2 Université Lyon 1, CNRS, LASIM, 43 bd du 11 Novembre 1918, Villeurbanne cedex, France

2 SUPPORTING INFORMATION Part I. Free energy calculation. The temperature dependence of the Helmholtz free energy, F, for a nanoparticle can be calculated using the harmonic approximation with the equation 1 : 3N 6 3N 6 F(T) = U hν 2 j + k B T ln 1 e hν j j=1 j=1 k B T where ν j are the vibrational frequencies of the nanoparticle and U 0 is the total energy of the optimized nanoparticle structure. Figure. SI-1 shows the calculated Helmholtz free energy as a function of temperature for different shapes of Au nanoparticles with ~ 300 and ~ 900 atoms. These results indicate that the energy order between different shapes obtained at T= 0 K remains up to room temperature since no crossings between the free energy curves were observed. Figure SI-1. Temperature dependence of the free energy for Au nanoparticles with TOC (red), DEC (yellow), and ICO (blue) shapes. (a) Au 928, Au 906 Au 923 ; (b) Au 314, Au 318, Au 309.

3 Part II. Continuous elastic theory: linear relations for the periods of the breathing and acoustic gap modes. For the breathing mode of a homogeneous isotropic sphere, the linear relation between the period of oscillation, τ BM, and the diameter, d, is given by: τ BM = πd η 0 c l, where c l is the longitudinal sound velocity, η 0 is the first positive root of the equation, tan (η) η = 1, 1 (ηc l 2c t ) 2 and c t is the transverse sound velocity 2. It should be noticed that the first positive root, η 0, depends on the material since the ratio of the longitudinal and transverse sound velocities are used to find η 0. 2 For the acoustic gap, it was found by Narvaez et al. 3, that it corresponds to a torsional oscillation of the homogeneous isotropic sphere. The equation which linearly relates this period of oscillation, τ AG, and the diameter, d, is, τ AG = πd ξ 0 c t, where c t is the transverse sound velocity, and ξ 0 is the first positive root of the equation, j 2 (ξ) 1 ξ j 2(ξ) = 0, which has the approximate value of j n (x) are the spherical Bessel functions of the first kind. In this case, ξ 0 does not depend on the material, in contrast with η 0. The sound velocities can be calculated from the measured bulk values of the Young modulus, E, the Poisson s ratio, ν, and the density, ρ, using the relations, 2,5 c l = 1 ρ K μ c t = μ ρ and for the bulk modulus K, and shear modulus, μ, K = E 3(1 2ν)

4 μ = E 2(1+ν). By using the tabulated values for E, ρ and ν, 6,7 the slope of the linear relations for the two periods can be calculated. Because of the discrepancies between different reported data for those elastic constants, we have used two sets of published values (References 6 and 7). The calculated values are shown in Tables SI-1 and SI-2. Table SI-1. The values for the Young modulus, E, density, ρ, and shear modulus, ν, were taken from Ref.[6]. The other tabulated parameters were calculated using the above equations. The slopes are given by C1 BM =π/η 0 c l and C1 AG-bulk =π/ξ 0 c t. ρ(kg/m 3 ) E(GPa) ν K(GPa) μ(gpa) c l (m/s) c t (m/s) η 0 C1 BM (ps/nm) C1 AG-bulk (ps/nm) Au Pt Ag Table SI-2. The values for the Young modulus, E, density, ρ, and shear modulus, ν, were taken from Ref.[7]. The other tabulated parameters were calculated using the above equations. The slopes are given by C2 BM =π/η 0 c l and C2 AG =π/ξ 0 c t. ρ(kg/m 3 ) E(GPa) ν K(GPa) μ(gpa) c l (m/s) c t (m/s) η 0 C2 BM (ps/nm) C2 AG-bulk (ps/nm) Au Pt Ag Part III.- Calculation of the quasi-breathing mode (QBM) period. To localize the eigenvector corresponding to the QBM, we use the procedure proposed by several authors, 8-10 which consists in calculating the projection of a homogeneous radial motion (HRM) into the entire set of eigenvectors {e i }. The HRM can be represented by the normalized vector positions of the nanoparticle, Then, using the equation, x = X X.

5 P i = x e i, to calculate the projection of the vector position over the whole set of eigenvectors, the QBM can be chosen as the one that maximizes the value of the projection. Figure SI-2 shows the value of P i as a function of the frequency for Au nanoparticles with ~ 900 atoms for the three structural families. Figure SI-2. Projections, P i, used to locate the QBM for Au nanoparticles.

6 Part IV.- Complete author list of Reference 28. Perez, A.; Mélinon, P.; Dupuis, V.; Bardotti, L.; Masenelli, B.; Tournus, F.; Prével, B.; Tuaillon- Combes, J.; Bernstein, E.; Tamion, A.; Blanc, N.; Taïnoff, D.; Boisron, O.; Guiraud, G.; Broyer, M.; Pellarin, M.; Del Fatti, N.; Vallée, F.; Cottancin, E.; Lermé, J.; Vialle, J. L.; Bonnet, C.; Maioli, P.; Crut, A.; Clavier, C.; Rousset, J. L.; Morfin, M. Int. J. of Nanotechnology 2010, 7, References [1] Landau, L. D.; Lifshitz, E. M. Statistical Physics; Pergamon: New York, 1980, Pt. 1. [2] Landau, L. D.; Lifshitz, E. M. Theory of Elasticity; Pergamon: New York, [3] Narvaez, G. A.; Kim, J.; Wilkins, J. W. Phys. Rev. B 2005, 72, (1-4). [4] Patton, K. R.; Geller, M. R. Phys. Rev. B 2003, 67, (1-10). [5] Sóliom, J. Fundamentals of the Physics of Solids I: Structure and Dynamics; Springer: Berlin, [6] Ledbetter, H. Monocrystal Elastic Constants and Derived Properties of the Cubic and the Hexagonal Elements: in Handbook of Elastic Properties of Solids, Liquids, and Gases, Vol. 2; Academic Press: New York, [7] Kaye, G. W. C.; Laby, T. H. Tables of Physical and Chemical Constants; Longman: 16th Edition, [8] Aguilar, J. G.; Mañanes, A.; Duque, F.; López, M. J.; Iñiguez, M. P.; Alonso, J. A. Int. J. Quantum Chem. 1997, 61, [9] Henning, C.; Fujioka, K.; Ludwig, P.; Piel, A.; Melzer, A.; Bonitz, M. Phys. Rev. Lett. 2008, 101, (1-4). [10] Combe, N.; Saviot, L. Phys. Rev. B 2009, 80, (1-7).