Enhanced cutoff energies for direct and rescattered strong-field photoelectron emission of plasmonic nanoparticles

2.1 Experimental setup

The laser system and VMI electron detection apparatus at the James R. Macdonald Laboratory at Kansas State University are described in more detail in [26, 55]. Briefly, the experiments used a Ti:Sapphire-based chirped pulse amplification (CPA) system generating 25 fs pulses FWHIM (10 optical cycles), and central angular frequency ω = 2.415 PHz (corresponding to a central wavelength λ = 780 nm). The laser pulse intersects the stream of isolated single NPs with diameters of 5, 30, or 70 nm that are injected by aerodynamic lens focusing [28, 55], [56], [57], [58]. As shown in the sketch of the experimental setup in Figure 1, PEs are projected onto the detector by the static electric field between the repeller and extractor. This allows the recording of the 2D projection of the PE momentum distributions as VMI maps. PE spectra were captured in a thick-lens, high-energy VMI spectrometer [59] capable of gathering up to 350 eV electron energy. The NPs were purchased from Cytodiagnostics [60]. The NP samples were custom synthesized, characterized for monodispersity (typical polydispersity index < 0.1) and sphericity (>95 %) to ensure sufficient reproducibility between interactions, and extensively purified to remove any source of contamination. We carefully chose the initial NP concentration to avoid the formation of clusters in the NP beam [61].

Figure 1:

Schematic of the velocity-map-imaging spectrometer coupled to the nanoparticle source. The dilute beam of isolated gas-phase nanoparticles is injected into vacuum and focused by an aerodynamic lens to intersect 800 nm, 25 fs, 10 kHz-repetition-rate linearly polarized laser pulses. Emitted electrons are focused onto the microchannel plate (MCP)/phosphor assembly. V_R and V_E are the respective voltages on the PE repeller and extractor plates needed to guide photoelectrons to the MCP phosphorus detector. The MCP is coupled to a phosphor screen, of which a camera records the spatial distribution of photoelectron hits for every laser shot.

2.2 Laser-intensity characterization

The peak laser intensity was determined by analyzing the above-threshold-ionization (ATI) PE energy distribution from gaseous Xe atoms with the VMI spectrometer described above and for the same laser parameters we selected for the strong-field-ionization studies reported in this work. To determine the absolute value of the intensity, the ponderomotive shift of the Xe ATI comb was measured as a function of the input-laser-pulse energy. From this shift, we deduced the ponderomotive energy, U_P, for a given pulse energy. Since U_p is proportional to the peak laser intensity I₀, the latter could be directly determined from this measurement. We determined the values of the intensities used in this work as I₀ = 8.0 × 10¹² W/cm² and 1.5 I₀ and estimated the accuracy of the intensity calibration to be better than 15 % (see Ref. [55, 62] for details).

2.3 Theoretical model

We numerically investigated PE emission from metallic NPs by IR-laser pulses with a Gaussian temporal profile. Propagating along the x axis and linearly polarized along the z axis, their electric field is given by

where τ is the pulse length at full-width-half-intensity maximum (FWHIM), ω the pulse’s central frequency, φ the carrier-envelope phase, and c the speed of light in vacuum (Figure 1). During the laser – NP interaction, LSPs are excited and induce an inhomogeneous plasmonic field near the NP surface. At the same time, and most significantly at the LSP resonance frequency [63, 64], electrons are excited to electronic states above the Fermi level. Sufficiently high laser intensities generate multiply ionized NPs [26, 56].

The incident laser pulse induces a transient dipole in the NP. Within the electric-dipole approximation, the corresponding transient induced plasmonic-dipole moment, P ⃗ pl ( t ) = ϵ 0 α Mie ( ω ) E ⃗ inc ( r ⃗ , t ) , generates the plasmonic electric field [65]

where k = 2π/λ = ω/c. We calculate the complex NP polarizability, α_Mie(ω), within Mie theory [66], following Ref. [67], which restricts the applicability of Eq. (2) to size parameters ka ⪅ 0.6 for nanospheres of radius a [68].

We describe strong-field ionization from metal NPs by extending the semi-classical three-step model (also known as “simple-man model”) for atomic strong-field ionization to metal NPs [17]. Our extended three-step model consists of: (1) electron release based on quantum-mechanical tunneling, (2) PE propagation from the NP surface to the detector by sampling over classical trajectories, and (3) PE rescattering and recombination at the NP surface. In comparison with gaseous atomic targets, each of these steps is significantly more intricate for metal NPs, due to their more complex electronic structure, the added morphological structure, and the emission of a much larger number of electrons, emphasizing the effects of PE – PE correlation, residual charges, and PE – nanoplasmonic-field interactions.

We represent the NPs’ static electronic structure in terms of the surface-potential step V₀ = ɛ _F + φ with the work function φ = 5.1 eV and Fermi energy ɛ _F = 8.0 eV for bulk gold [69]. Our dynamical numerical simulation divides the NP surface into small surface elements. During successive small time intervals, the surface elements are modeled as spherical square-well potentials. Bound PEs close to the NP surface tunnel out along the radial component of the total electric field at the NP surface, F ⃗ ⋅ e ̂ r , where F ⃗ = E ⃗ inc + E ⃗ pl + F ⃗ res . The residual-charge field F ⃗ res results from the accumulation of positive residual charge on the NP during electron emission in preceding time intervals. We account for strong-field electron release from the NP by employing modified [17] Fowler–Nordheim tunneling rates [70, 71]. Subsequently, we Monte Carlo sample over the initial phase-space distribution of released electrons and solve Newton’s equations of motion for the PE propagation outside the NP in the presence of all electric fields, F ⃗ + F ⃗ e − e , where F ⃗ e − e is the repulsive Coulomb electric field between PEs. In each laser half-cycle the direction of the incident-laser electric field changes, such that emitted PEs can be driven back toward the NP and either rescatter from or recombine at the NP surface. For 5, 30, and 70 nm diameter gold nanospheres, we include and numerically evaluate the effects of PE repulsion, residual positive charges on the NP, PE recollisions and recombinations at the NP surface, and nanoplasmonic enhancement of the incident-laser-pulse electric field. More details about this numerical model are given in the Supplementary Information (SI) and in Ref. [17].

In our numerical applications in Section 3, we distinguish and compare specular and diffusive PE rescattering at the NP surface. For diffusive rescattering, we uniformly randomize the polar and azimuthal scattering angles relative to the surface normal at the impact site on the NP surface, modeling rescattering in all accessible directions with equal probability.

3.1 Influence of nanoplasmonic field, rescattering, residual-charge interactions, and photoelectron correlation

VMI spectra are sensitive to all PE interactions included in our simulation. In order to track the effects of different electronic interactions on the propagation and rescattering of released PEs, we leave the modeling of the tunneling release of electrons at the NP surface unchanged when selectively switching off individual PE interactions (for identical laser-pulse parameters), assuming for all calculated VMI maps identical tunneling-ionization rates (Eq. (S1.5) in the Supplementary Information) The comparison of simulations in which we selectively include and exclude specific PE interactions during the PE propagation and rescattering, allows us to quantify their specific effects on VMI maps.

Figure 2 shows simulated VMI spectra compared to experimental results for gold nanospheres with 30 nm diameter for the experimental setup depicted in Figure 1. The VMI spectra are projections of the PE momentum distribution on the x–z plane of the MCP detector and show the projected PE yields as functions of the PE asymptotic velocities, v _x and v _z , along the laser-propagation and -polarization directions. The first, second, and third column in Figure 2 include, respectively, VMI spectra of direct PEs, rescattered PEs, and the net PE yield for either specular (first and second row) or diffuse rescattering (third and fourth row, cf., Sec. (S3) in the SI). The first and third row show simulations in which only the incident-laser and plasmon fields ( E ⃗ inc and E ⃗ pl ) are considered. The VMI spectra in the second and fourth row include all forces: E ⃗ inc , E ⃗ pl , PE interactions with residual positive charges ( F ⃗ res ) , and repulsive PE Coulomb interactions ( F ⃗ e − e ) . Figure 2(n) is our measured VMI map for the same laser and NP parameters.

Figure 2:

(a–m) Photoelectron VMI spectra simulated for 30 nm diameter gold nanospheres for direct (first column), rescattered (second column), and all (denoted as “Net”, third column) photoelectrons, including either specular (first and second row) or diffuse rescattering (third and fourth row) for incident 780 nm laser pulses with a pulse length of 25 fs (FWHIM) and 8.0 × 10¹² W/cm² peak intensity. μ designates the integrated yield, normalized to the simulated net PE yield. First and third row: simulations where only the incident-laser and plasmon fields ( E ⃗ inc and E ⃗ pl ) are included. Second and fourth row: VMI spectra including all the interactions, E ⃗ inc , E ⃗ pl , photoelectron interactions with residual positive charges ( F ⃗ res ) , and repulsive photoelectron Coulomb interactions ( F ⃗ e − e ) . (n) Corresponding measured VMI spectrum.

To allow for a quantitative comparison of direct and rescattered PE yields, we normalized the yields in each row to the corresponding net PE yield in the third column and display the normalized integrated yield μ in each graph of Figure 2. We calculated μ as the v _x - and v _z -integrated yields from the simulated VMI maps in each row, divided by the corresponding integrated yield of the VMI maps in the third column. The comparison of the VMI spectra in Figure 2 allows us to assess the influence of the distinct PE interactions on VMI spectra, as we discuss next.

3.2 Influence of nanoparticle size and laser intensity

Figure 3 shows simulated and experimental VMI spectra for gold nanospheres with diameters of 5, 30, and 70 nm. The first, second, and third column are simulated VMI spectra for, respectively, the direct, rescattered, and net PEs yield for peak laser intensities I₀ (first, second, and third row) and 1.5 I₀ (fourth, fifth, and sixth row). Experimental results corresponding to the simulations in the third column are shown in the fourth column. The VMI spectra in Figure 3 are (slightly) elongated along the laser-polarization direction, with PE cutoff energies that increase with NP size. As discussed in Section 3.1, isotropic VMI spectra are promoted by PE–PE interactions and diffuse PE rescattering from the NP surface, while incident-laser and induced plasmonic-field interactions tend to imprint a dipolar shape. The detected number of the PEs per laser short for the experimental data shown in Figure 3 varies from 140 for 5 nm diameter NPs at the lower peak laser intensity (I₀) to 600 for 70 nm NPs at 1.5 I₀. However, as discussed in detail in Ref. [55], these numbers do not directly reflect the number of PEs that hit the detector due to the PE energy-dependent detector saturation in our experiment. The saturation effect is most prominent in the central detection area, where low-energy electrons (which dominate the total PE yield) hit the MCP.

Figure 3:

Comparison of simulated direct (first column), rescattered (second column) and net (i.e., including direct and rescattered yields, third column) photoelectron VMI spectra with experimental (forth column) VMI spectra for gold nanospheres with 5, 30, and 70 nm diameter and laser peak intensities of I₀ = 8.0 × 10¹² W/cm² (first – third row) and 1.5 I₀ (forth – sixth row). The laser-pulse length and wavelength are 25 fs and 780 nm. Red dashed circles in (a–x) indicate simulated and experimental photoelectron cutoff energies. μ is the integrated photoelectron yield normalized to the integrated net yields in third column.

To allow for a quantitative comparison of direct and rescattered PE yields, we normalized the direct and rescattered PE yields in each row to the corresponding net PE yield in the third column and displayed the normalized integrated yield μ(a, I₀) in each graph. μ reveals that the yield of direct PEs decreases as a function of the NP size and intensity, being more sensitive to the size. This observation is compatible with PEs having a higher probability to rescatter off larger NPs. In addition, increasing laser intensity leads to a stronger radial attractive force, due to an increase in the number of residual charges on the NP surface, leading to more PE rescattering events. The direct and rescattered PE yield can be controlled by the intensity of the laser pulse and size of the NP. The measured and simulated VMI maps also reveal a large increase in the direct and rescattered PE cutoff energy with the laser peak intensity and NP size. We quantify this laser-intensity and NP-size-dependent effect in the following subsection.

3.3 Angle-integrated photoelectron yields and cutoff energies

Figure 4 shows simulations corresponding to the VMI spectra in Figure 3. It includes (i) all interactions for the direct PE yield (denoted as “All_Direct”), (ii) all interactions for the rescattered PE yield (“All_Rescat”), and (iii) all interactions for the net PE yield (“All_Net”). In addition, Figure 4 displays (iv) simulations only including incident- and plasmonic-field interactions for the net PE yield (denoted as “Inc + Pl_Net”) and (v) integrated experimental yields as a function of the PE kinetic energy. Due to the detector saturation at the center of the MCP phosphor detector (Figure 1), the experimental yields for kinetic energies below approximately 8 eV (corresponding to PE velocities below 0.8 a.u.) are not accurate. To be able to compare experimental integrated yields to one another and to the simulation results, we have removed the low energy part of the integrated yields from both, experimental and simulated data.

Figure 4:

Photoemission yields as functions of the photoelectron kinetic energy for gold nanospheres with (a) and (d) 5, (b) and (e) 30, and (c) and (f) 70 nm diameter and laser peak intensities of (a–c) I₀ = 8.0 × 10¹² W/cm² and (d–f) 1.5 I₀. The laser-pulse length and wavelength are 25  fs and 780 nm. Simulated photoelectron yields including all interactions are shown for directly emitted (“All_Direct”), rescattered (“All_Rescat”), and net (“All_Net”, i.e., direct and rescattered) photoelectrons. Simulations only including incident- and plasmonic-field interactions are denoted as “Inc + Pl_Net”. Black dots show experimental yields.

The overall agreement between experimental and simulated integrated PE yields in Figure 4 is not perfect for several reasons. With regard to the simulation, an important uncertainty derives from our implementation of approximate modified Fowler–Nordheim tunneling rates. With regard to the experiment, the above-mentioned detector saturation decreases the reliability of the low-energy portion of our spectra. While the low-energy portion of the simulation data was truncated to allow for a better comparison with the experiment, the detection uncertainty due to saturation is not completely removed and tends to affect predominantly our measurements for the largest NP size (70 nm diameter) and the higher laser peak intensity (1.5 I₀), due to larger numbers of emitted PEs per laser shot. This is consistent with the agreement between simulation and measurement being better for 30 nm NPs at the lower peak intensity (I₀) in Figure 4 than for 70 nm NPs in Figure 4(c) and at the higher laser intensity of 1.5 I₀ in Figure 4(d)−(f). However, in view of hardly avoidable inaccuracies in the detailed modeling of this complex interaction scenario and NP-size- and laser-intensity-dependent experimental background noise, we cannot exclude that the exceptionally good match between experimental simulated results shown in 4(b) compared to the other graphs in this figure is serendipitous.

Integration of the VMI-projected PE momentum distributions y(v _x , v _z ) in Figure 3(a)–(x) over the PE detection angle ϕ in the VMI plane results in PE yields

as functions of the PE energy, E PE = v x 2 + v z 2 / 2 . The yields Y(E_PE) shown in Figure 4(a)–(f) are normalized individually to their maxima, except for the simulations labeled “Direct_Net” and “Rescat_Net”, which are normalized to the maxima of the “All_Net” simulation results.

For simulated yields, we define the PE cutoff energy E_cutoff as the energy up to which 99.5 % of the net PE yield has accumulated,

The experimental cutoff energy was extracted from the experimental VMI maps as described in [26, 55], for which the upper energy boundaries of the full 3D momentum sphere and the 2D projection are identical. The radial distribution of these projections along the polarization direction accurately determines the maximum PE energy.

The PE cutoff energies, shown as red dashed circles in Figure 3 in Section 3.2, increase with the NP size and peak laser intensity. Figure 5(a) and (b) display cutoff energies as a function of NP size for peak intensities of I₀ = 8.0 × 10¹² W/cm² and 1.5 I₀, in units of the incident-laser ponderomotive energies U_p(I₀) and U_p(1.5 I₀), respectively. Blue diamonds and red circles show, respectively, simulated cutoff energies (including all interactions) for the direct (denoted as “All_Direct”) and net (“All_Net”) PE yields. Gray squares with error bars are experimental cutoff energies (“Experiment”). Simulation results including all interactions for rescattered PEs are not shown, because they coincide with the “All_Net” yield. For gaseous atomic targets, the cutoff energy is equal to 10 U_p [30–33]. Cutoff energies obtained by scaling this well-known expression by the plasmonic intensity enhancement of the incident-laser pulse, η², are shown as yellow “plus” markers. As expected, they tend to merge with the cutoff energies computed while only including incident-laser-pulse and plasmonic-field interactions (represented as green triangles). We calculated the applied value for η within Mie theory at the poles (relative to the laser-polarization direction) of the NPs [66–68]. In contrast to the 10 η²U_p scaling, the comparison of Figure 5(a) and (b), shows that our theoretical cutoff energies predict intensity-dependent changes that become more pronounced for larger NPs. Within the experimental error this theoretical prediction is compatible with our experimental results.

Figure 5:

Comparison of simulated and experimental photoelectron cutoff energies scaled by the incident-laser ponderomotive energy U_p(I₀) for 5, 30, and 70 nm diameter gold nanospheres and laser peak intensities of (a) I₀ = 8.0 × 10¹² W/cm² and (b) 1.5 I₀. The laser-pulse length and wavelength are the same as in Figure 4. Simulated cutoff energies including all interactions for, respectively, direct (“All_Direct”) and net (“All_Net”, i.e., direct and rescattered) photoemission. Simulations only including incident- and plasmonic-field interactions are denoted as “Inc + Pl_Net”. Yellow “plus” markers show atomic cutoff energies, 10 U_p, scaled by the plasmonic intensity enhancement η².

Based on the discussion in Section 3.1 of different PE interactions and their influence on VMI maps, we investigated two plausible causes for the numerically predicted increase of the PE yield and cutoff energy with the NP size. First, the lowering and narrowing of the surface-potential barrier by the more significant nanoplasmonic-field enhancement near larger NPs [12, 18, 21, 68] promotes strong-field tunneling ionization. However, this not only tends to augment the measured PE yield. Since PEs gain more energy in a more strongly enhanced field, it also entails higher cutoff energies for larger NPs. Second, as the NP size increases, a larger surface area becomes available from where more electrons are emitted, increasing the PE yield. The cutoff energy rises with the PE yield due to the increased repulsive Coulomb energy between PEs upon their release from the NP surface. In principle, a third cause for larger yields and cutoff energies can be laser-pulse-propagation effects inside the NP that result in higher local-field enhancements for larger NPs [56]. However, for the NP sizes investigated here, we did not find this effect to be relevant.

As discussed in Section 3.1, the consequences of residual-charge interactions and PE–PE interactions oppose each other. While attractive residual-charge – PE interactions reduce both, PE yields and cutoff energies, PE Coulomb repulsion increases them. A detailed numerical comparison of these competing interactions is shown in Figure 3 of Ref. [17]. Our numerical results indicate that PE Coulomb repulsion overcompensates residual-charge – PE interactions with regard to the cutoff energy, leading to an overall cutoff-energy increase, especially for larger NPs.

The green triangles (denoted as “Inc + Pl_Net”) in Figure 5 are cutoff energies calculated under the assumption that released electrons solely interact with the incident-laser and induced plasmonic field while propagating to the detector. In Sec. (S4) of the SI, we derive a closed-form analytical heuristic expression for the cutoff energy in direct photoemission, based on a simplified central-field approximation of residual-charge interactions and PE correlation. By comparing Eq. (S4.12) in the Sec. (S4) with the known respective 2 U_p and 10 U_P limits for direct and rescattered emission for atomic strong-field ionization, we infer the cutoff energy for rescattered PEs,

where η eff R ( a , I 0 ) models, on average, the effect of plasmonic-field enhancement on rescattered PEs (indicated by the superscript “R”), while taking all PE interactions into account. t _f designates the effective interaction time (determined at numerical convergence). We introduce the effective Coulomb interaction factor, η C R ( a , I 0 ) = η e − e R ( a , I 0 ) − η res R ( a , I 0 ) , in analogy to the plasmonic-field-enhancement factor η. In the central-field approximation, η e − e R ( a , I 0 ) and η res R ( a , I 0 ) represent PE–PE repulsion and the decelerating effect of residual-charge interactions, respectively. Note that direct PEs, on average, are more strongly affected by repulsive PE–PE Coulomb interactions and plasmonic-field enhancement than rescattered electrons, as mentioned earlier. For the NP and laser parameters we considered, this leads to comparable cutoff energies for direct emission and rescattering. η C R ( a , I 0 ) is a measure for the magnitude of the effective counteracting attractive (decelerating) residual-charge and repulsive (accelerating) PE–PE interactions. For the present work η C R ( a , I 0 ) > 0 , indicating the dominance of PE repulsion over residual-charge attraction in determining the cutoff energy.

The heuristic Eq. (5) qualitatively explains all experimental results in Figure 5. For very small η C R ( a , I 0 ) and plasmonic-field enhancement, Eq. (5) approaches the familiar 10 U_p(I₀) scaling for rescattering ionization of gaseous atomic targets, as expected. This condition is satisfied for dielectric NPs for appropriate particles sizes and laser intensities. Including field enhancement simply in terms of a multiplicative factor, η², i.e., modeling cutoff energies as 10 η²U_p(I₀), fails to reproduce the cutoff energies we measured and simulated for metal NPs when plasmonic-field enhancement and PE Coulomb interactions are relevant. If η C R ( a , I 0 ) is negligible, the cutoff energy, 10 η eff R ( a , I 0 ) 2 U p ( I 0 ) , is smaller than 10 η²U_p(I₀), since η eff R ( a , I 0 ) < η , (see Sec. (S4) in the SI). For η C R ( a , I 0 ) > 0 , PE correlation dominates the residual-charge deceleration, and cutoff energies tend to rapidly increase with the NP size and laser intensity. For the present numerical applications, even small η C R ( a , I 0 ) enable very large cutoff energies, because the coefficient ωt _f in Eq. (5) is large. Assuming an effective PE interaction time t _f ≈ 2τ, for the laser parameters used in this study, ωt _f = 120.75.

An unexpected and interesting result derives from the fact that the simulated cutoff energies for direct PEs (“All_Direct”) and rescattered PEs (“All_Net”) are comparable. The direct cutoff energy for 5, 30, and 70 nm NPs are, respectively, 93, 85, and 89 % of the rescattered PE cutoffs for the intensity I₀ and 93, 87, and 84 % for 1.5 I₀. This value is 20 % for strong-field ionization of gaseous atoms, molecules, and dielectric NPs.