The role of morphology and coupling of gold nanoparticles in optical breakdown during picosecond pulse exposures

This paper presents a theoretical study of the interaction of a 6 ps laser pulse with uncoupled and plasmon-coupled gold nanoparticles. We show how the one-dimensional assembly of particles affects the optical breakdown threshold of its surroundings. For this purpose we used a fully coupled electromagnetic, thermodynamic and plasma dynamics model for a laser pulse interaction with gold nanospheres, nanorods and assemblies, which was solved using the finite element method. The thresholds of optical breakdown for off- and on-resonance irradiated gold nanosphere monomers were compared against nanosphere dimers, trimers, and gold nanorods with the same overall size and aspect ratio. The optical breakdown thresholds had a stronger dependence on the optical near-field enhancement than on the mass or absorption cross-section of the nanostructure. These findings can be used to advance the nanoparticle-based nanoscale manipulation of matter.

whereε, is the size-dependent dielectric constant for gold. It is given by: S1 (ω, L eff ) =˜ bulk + ω 2 p ω 2 + iωγ 0 − ω 2 p ω 2 + iω γ 0 + Av F L eff + ηVau π . (2) The incident light was polarized along the y-axis (the longest axis of the nanostructure) and travelled along the positive z-direction. Its electric field was, E inc : where κ w is the wave number in medium. The dielectric function of water was modelled using Drude formalism, to account for a presents of free electron plasma as: where ε ∞ is the water relative permittivity (assuming biologically relative refractive index of 1.4), m is the electron reduced mass and τ is the mean free time between electron/molecule collisions.
The size-dependent corrections to the complex dielectric function of gold,ε,are based on the recent quantitative comparison of a finite element model of a single gold nanorod against spatial modulation spectroscopy (SMS) technique. S1 The solution to the Helmholtz wave equation provides the optical behaviour of a gold nanoparticle, such as electric field enhancement, |E|/E 0 and absorption cross-section, σ abs . The combination of perfect matched layer (PML) domain and absorbing boundary condition was used to truncate electromagnetic simulations domain and reduce reflections from artificial boundaries. The perfect electric (PEC) and perfect magnetic (PMC) conductor boundaries were used to truncate domain to one quarter of full 3D geometry.

Two temperature model
The resistive losses during laser pulse interaction with gold nanoparticle, Q rh , calculated from the electromagnetic model is fed into a hyperbolic two-temperature model (TTM) S2 to solve for the energy transfer between the laser pulse and the conduction electrons of gold due to electron-phonon relaxation and heat diffusion from the gold lattice to the surrounding medium through interface conductance, Q au|w : S3

Plasma formation
The temperature rise of the conduction electrons in gold from the TTM and the electric field distribution from the electromagnetic model are used as input parameters to the plasma model. The interaction of the strong electromagnetic fields with the electrons in a aqueous medium can lead to multiphoton, impact and avalanche ionization of the electrons into a quasi-free state in the conduction band. When the density of the excited electrons reaches a critical value of 10 18 −10 20 cm −3 , S4-S9 the electron cloud starts to gain sufficient kinetic energy from the interaction with the laser pulse that an optical breakdown occurs. Laser-induced S3 breakdown can lead to the breakage of atomic bounds, bubble formation, luminescence and acoustic shock formation. In order to determine threshold of plasma induced optical breakdown in aqueous media, the medium can be represented as water S10,S11 with properties of an amorphous semiconductor with a band gap energy of E gap = 6.5 eV. S7,S12,S13 To model the changes in free electron density and determine the irradiance threshold for optical breakdown, the generic form of the rate equation for free electron density, ρ e , can be used: S11 The dρe dt photo term on the right-hand side of Equation (10), models photoionization of electrons via multiphoton absorption and tunnel ionization. S14-S16 The dρe dt casc term adds the contribution of cascade ionization (sometimes called avalanche ionization) via inverse Bremsstrahlung absorption. S17-S19 The dρe dt diff and dρe dt rec terms represent free electron plasma diffusion and recombination, respectively.
To complete the picture of the free electron plasma formation by laser pulse interactions with a gold nanoparticle, two additional processes needs to be considered: thermal ionization (thermionic emission), S20,S21 ρ therm , and photo-thermal emission, S22 ρ au .
The thermal ionization starts to play a significant role in free electron production when the temperature in the focal volume of the laser is sufficiently high (≈ 5000 K and above S20,S21 ) and large free electron densities are already achieved by multiphoton and cascade ionization.
On the other hand, thermal ionization also plays a role during sequences of ultrashort laser pulses, when the time delay between pulses is in the order of thermalization time of the electron plasma energy. S21 Thermal ionization also partially depletes the density of bound electrons in the valence band and reduces the rate of multiphoton and impact ionization. S20 Although in the model, the temperature of the medium during single pulse illumination is sufficiently lower than 5000 K, we incorporated thermal ionization therm, based on derivations given by Linz et al., S20 into free electron density rate Equation (10) to have complete picture of the free electron plasma generation, so that it can be used for future studies of S4 ultrashort pulses of a high repetition rates. Please note that in present single 6 ps pulse study thermal ionization of the water is negligible and can be omitted.
Photo-thermal emission of hot electrons on the boundary of the gold, also starts to play a role when the temperature of the nanoparticle's electrons rises to a level where the electrons can cross the metal/medium energetic barrier of W au = 3.72 eV and contribute to electron density plasma formation. S23 4 terms in the generic rate Equation (10), plus thermal ionization of water, ρ therm , and photo-thermal emitted electron current density across gold/water boundary, of low density plasma formation are expanded as follows: The photoionization rate, S20 dρe dt photo : where γ is the Keldysh parameter given by: here, 1/ω t , is the tunnelling time through the atomic potential barrier, m is the electron reduced mass, and∆ is the effective ionization potential for creating an electron-hole pair S5 in a condensed matter exhibiting a band structure and corrected for the oscillation energy of the electron due to the electromagnetic field. It is given by: In Equation (11), < x > represents the integer part of the number x, K() and E() denote the elliptic integrals of the first and second kind, and Φ() denotes the Dawson probability integral: The cascade ionization rate, S20 dρe dt casc Cascade ionization contributes to the rate Equation (10) only after certain initial free electron density, ρ seed , has been reached by either photoionization or photo-thermal emission.
The initial seed free electron density is the fitting parameter and is chosen based on 50 % probability of having at least one seed electron in the focal volume of laser. S20 Then cascade ionization rate can be written as: Thermal ionization, S20 ρ therm Based on derivation of Linz et al., S20 the third process of free electron generation can come from thermalization of energy, carried by primary free electrons via thermal emission.
Thermally ionized free electron density in the conduction band is calculated by: Photo-thermal emission current, J au The photo-thermal emitted electron current density across metal/medium boundary can be described by generalized Fowler-DuBridge theory of multiphoton photoemission at high temperatures: S24,S25 where A 0 is the Richardson coefficient (120 A/cm 2 /K 2 ), W au is work function (3.72 eV), S23 F is the so called Fowler function, S24,S25 c au is the three-photon ionization cross-section (1 × 10 −7 Acm 4 /MW 3 ), S26 κ b is the Boltzmann constant and R is the reflection coefficient of the gold. The photo-thermal electron current density is set as a boundary flux/source condition across gold/water interface.
The details for diffusion, dρc dt diff , and electron-hole recombination rates, (η rec ) of the Equation (10) are given in the Table S1. The first one is based on the characteristic diffusion length, Λ, which is set to the radius of the gold nanoparticle. S27 The electron-hole recombination rate is set to empirical value obtained by Docchio. S28 By implementing above mentioned terms into generic rate Equation (10), plus introducing photo-thermal emission current as a boundary condition, leads to a full description of free electron plasma formation in the vicinity of gold nanoparticle in the form: The transient changes in the optical behaviour of the s25t@640 nanoparticle with a fluence of 0.37 mJ/cm 2 are shown in Figure S1. The dark red line in Figure S1 represents the tempo-S7 ral evolution of maximum electric field enhancement in the middle of 4 nm gap between two adjacent nanospheres. As the density of free electron plasma, ρ e , reaches 1×10 19 cm −3 the plasma becomes highly absorbent to incident laser irradiation and shields the gold nanoparticle. The maximum field enhancement reaches its minimum at the maximum density of the plasma (dip in the near-field enhancement, Figure S1). The 4 ps difference between peak of the laser intensity and the peak of the free electron density is caused by a domination of photo-thermal emission of hot electrons off the nanoparticle's surface, where photo-thermal emission is proportional to T 2 e I 3 tot .  Figure S1: Temporal evaluation free electron plasma, ρ e (cm −3 ), shown by the green line; and maximum electric field enhancement, |E| max /E 0 , shown by the dark red line for 25 nm nanosphere trimer (s25t@640) irradiated at the resonance peak wavelength, 640 nm, with the fluence of 0.37 mJ/cm 3 . The probe point located at the middle of the edge-to-edge gap between two adjacent nanoparticles. The grey shaded area shows the laser intensity profile (in arbitrary units). The time is normalized to FWHM of the laser pulse, τ w .

Heat transfer in aqueous media
The temperature increase in the aqueous media, T w , due to the laser-particle interaction, linear absorption (Joule heating), plasma formation and optical breakdown, is modelled by solving heat transfer equation. There were four types of heat sources: 1) a Dirichlet boundary condition of the interface conductance at the nanoparticle surface -Equation (9); 2) Joule heating with the plasma produced in the vicinity of the nanoparticle, Q rh ; 3) electron collision losses with neutral molecules during impact ionization, dTm dt coll ; 4) heating through electron recombination of ionized molecules, dTm dt rec . With above mentioned considerations, the heat transfer equation will following: where heat source, Q, is:  κ e see ref. S35 Electron thermal conductivity κ l see ref. S22

Lattice thermal conductivity
Continued on next page S10 ρ au see eq. 18 Density of thermally ionized electrons in the medium S20,S21 ρ therm see eq. 17 Density of electrons emitted off the gold surface S25 c w 4184 J/kg/K Heat capacity of water ρ w 1000 kg/m 3 Density of water κ w 0.61 W/m/K Thermal conductivity of water T w see eq. 19 Temperature of water q 0 105.0 × 10 6 W/m 2 /K Thermal conductance at gold-water interface S37