Generation of fast photoelectrons in strong-field emission from metal nanoparticles

2.1 Experimental setup

The experiments were carried out at the James R. Macdonald Laboratory at Kansas State University. The laser system, NP source, and VMI electron detection assembly is described in more detail in Refs. [39], [50]. Briefly, we used a Ti:Sapphire-based chirped pulse amplification (CPA) system generating 25 fs (10 optical cycles) full-width-half-intensity maximum (FWHIM) pulses with a central angular frequency of ω = 2.354 PHz (corresponding to a central wavelength of λ = 800 nm). The NP source consisted of an atomizer, dryer, and aerodynamic lens and injects single, isolated particles into the vacuum, where the beam of NPs intersects the focused intense laser beam [41], [50], [51], [52], [53]. As depicted in the sketch of the experimental setup Figure 1, PEs generated by strong-field ionization of NPs in the laser pulse, are projected onto the detector by the static electric field between the repeller and extractor of the VMI system, allowing for the recording of the 2D projection of the PE velocity distribution. The thick-lens, high-energy VMI spectrometer is capable of detecting PEs with kinetic energies up to 350 eV [54]. The colloid NP samples with low polydispersity and high purity were purchased from Cytodiagnostics Inc. (gold and silver NPs) and Nanocomposix (platinum NPs) [55], [56].

Figure 1:

(Color online) Schematic of the NP source and velocity-map-imaging spectrometer setup. A beam of single, isolated NPs is injected into vacuum by an aerodynamic lens and intersects an intense linearly polarized 800 nm, 26 fs laser beam operated at a repetition-rate of 10 kHz. Emitted electrons are focused onto a microchannel plate (MCP) and phosphor assembly, from which a camera records the spatial distribution of PE hits for individual laser shots.

2.2 Laser-intensity characterization

For the laser intensity and PE momentum calibration, we employed the VMI spectrometer described above to measure above-threshold-ionization (ATI) PE momentum distributions from gaseous Xe atoms [57]. This measurement was done under the same experimental conditions as for the photoemission-from-NP experiments reported here. From the 2D projection of the PE momentum distributions, we determined the ponderomotive shift of the ATI combs. Based on the proportionality of the ponderomotive energy and peak intensity of the incident laser pulse, I ₀, we determined the intensities used in this work in the interval 10¹² W/cm² < I ₀ < 3 × 10¹³ W/cm². We estimated the accuracy of the intensity calibration to be better than 15 % [50], [58].

2.3 Theoretical model

The high laser intensities we consider generate multiply ionized NPs [39], [51]. We numerically model 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 FWHIM, ω the pulses’ central frequency, ψ 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. Most significantly at the LSP resonance frequency [59], [60], electrons are excited to electronic states above the Fermi level. Sufficiently high laser intensities generate multiply ionized NPs [39], [51]. The incident laser pulse thus transiently polarizes the NP. Within the electric-dipole approximation, the corresponding transient induced plasmonic dipole moment, P ⃗ p l ( t ) = ε 0 α Mie ( ω ) E ⃗ inc ( r ⃗ , t ) , gives raise to the plasmonic electric field [61]

where k = 2π/λ = ω/c. We calculate the complex NP polarizability, α _Mie (ω), within Mie theory [62], [63], which restricts the applicability of Eq. (2) for nanospheres of radius a to size parameters ka⪅0.6 [64].

We describe strong-field ionization from metal NPs by extending the semi-classical three-step model (also known as the ‘simple-man model’) for atomic strong-field ionization to metal NPs [26]. 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. This is due to the NPs’ more complex electronic structure, the added morphological structure, and the emission of a much larger number of electrons compounding 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 V ₀ = ɛ _F + φ, given by the work function φ and Fermi energy ɛ _F for bulk metal [65]. During successive small time intervals, our dynamical numerical simulation divides the NP surface into small surface elements, which are modeled as square-well potentials. Bound PEs close to the NP surface are assumed to tunnel out in radial direction, driven by the radial component of the total electric field at the NP surface, F ⃗ ⋅ e ̂ r , where F ⃗ = E ⃗ inc + E ⃗ p l + 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 [26] Fowler-Nordheim tunneling rates [66]. 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 accelerated back toward the NP and either rescatter from or recombine at the NP surface. 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 the numerical model are given in Ref. [26] and the Supplementary Information of Ref. [49].

3.1 Photoelectron-momentum images

We start the presentation of our results with a comparison of laser-intensity and size-dependent measured and numerically simulated PE spectra for silver, gold, and platinum nanospheres. Figures 2, 3, and S1 display simulated and experimental VMI spectra for silver nanospheres with diameters of 10, 60, and 100 nm, respectively. The first, second, and third columns present simulated VMI spectra for the direct, rescattered, and net PE yields. The rows represent results for different peak laser intensities I ₀. Our numerical results in the third column are compared with our measured VMI spectra, shown in the fourth column for the three highest intensities. To facilitate a quantitative comparison between the direct and rescattered PE yields, we normalized the yields in each row to the corresponding net PE yield in the third column and display normalized integrated yields μ in all graphs.

For comparison with the simulated VMI spectra for silver NPs in Figures 2, 3, and S1, we provide in Figures S2, S3, and S4 of the Supplementary Information (SI) corresponding calculated spectra for gold NPs of identical diameter for the same peak intensities. Additional measured and simulated VMI spectra for gold NPs with different diameters and for other laser intensities than in the present work can be found in Ref. [49]. Figures S5 and S6 in the SI show a comparison of simulated and experimental VMI spectra for gold and platinum NPs with a diameter of 70 nm, respectively. For the examined transition metals, all simulated VMI spectra exhibit a slight elongation along the laser polarization direction, with PE cutoff energies and yields increasing with NP size and laser intensity. The high degree of isotropy is a direct result of the Coulomb repulsion between PEs and diffuse PE rescattering from the NP surface, as discussed for gold NPs in [26], [49].

We expect the yield parameter μ for rescattered PEs to consistently increase with both NP size and laser intensity: Higher laser intensities result in enhanced PE emission from the NP surface and higher residual charge accumulation on the NP. The accumulated residual charge enhances the yield of rescattered PEs. An increase in NP size provides a larger surface area for the laser - NP interaction, leading to increased PE emission and, consequently, enhanced PE rescattering. This effect is somewhat mitigated at larger sizes due to the, on average, larger distance between emitted PEs and accumulated residual charges.

The increase in yield with both NP size and intensity is nonlinear. An increasing residual charge on the NP tends to impede electron emission and, thus, to slow the growth of the electron yield with increasing laser intensity. This effect is comparable to the increase of the residual charge and work function of fullerene clusters during sequential electron capture by slow highly charged ionic projectiles [67], [68], [69]. Below, we shall refer to this observation as ‘yield-saturation effect’. It occurs with increasing laser peak intensity I ₀ at larger PE yields. We hold the underlying electronic dynamics accountable for the complex change of the branching ratio between direct and rescattered PE yields as a function of the laser intensity and NP size (cf., Figures 2, 3, S1, and SI).

At low intensities, the yield of rescattered PE increases relative to the direct-emission yield with the NP size, as is seen by comparing, e.g., Figures 2b and 3b. This yield increase is compatible with the larger effective surface area (geometrical cross section) for laser – NP interactions and electron rescattering. Consequently, an increase in NP size leads to higher PE yields and promotes rescattering. In contrast, the rescattering yield decreases for increasing NP size relative to direct emission at high intensities, as is illustrated by comparing, e.g., Figure 2(i) and 3(i) or Figure 2(m) and 3(m) for silver NPs. This indicates that now, at high intensity, the increase of the geometrical rescattering cross is less competitive in promoting rescattering over direct emission. The relative smaller chance for rescattering is compatible with the lower rate of residual-charge accumulation that is also deemed responsible for the yield-saturation (cf. Figure 5).

Figure 2:

VMI PE spectra for strong-field ionization of silver nanospheres with 10 nm diameter and laser peak intensities of 4.0 × 10¹², 7.0 × 10¹², 1.36 × 10¹³, and 2.65 × 10¹³ W/cm² (first - fourth row, respectively). Comparison of simulated direct (first column), rescattered (second column), and net (direct plus rescattered yields, third column) spectra with experimental spectra (fourth column). The laser-pulse length and wavelength are 25 fs FWHIM and 800 nm, respectively. In each row, calculated integrated PE yields for direct and rescattered emission, μ, are normalized to the integrated net yields in the third column. The red circle in each VMI map represents the photoemission cutoff. The cutoff energies are given in red above the VMI maps. No experimental data is available for 4.0 × 10¹² W/cm² peak intensity.

Figure 3:

As Figure 2 for 60 nm diameter silver nanospheres and laser peak intensities of 1.7 × 10¹², 4.0 × 10¹², 6.7 × 10¹², and 1.32 × 10¹³ W/cm² (first – fourth row, respectively).

3.2 Cutoff energies and net PE yields

In this subsection we discuss laser-intensity and NP-size dependent PE cutoff energies and total yields for silver, gold, and platinum nanospheres. Extended exposure to the laser electric field leads to rescattering as the dominant mechanism for reaching the highest observable PE kinetic energies. The PE cutoff energy E cutoff R is thus determined by rescattered PEs (denoted by the superscript ‘ R’). The effects of PE Coulomb repulsion, PE attraction by the built-up residual positive charge of the NP, and plasmonic field enhancement on E cutoff R can be represented by the heuristic formula [49].

This expression describes the influence on photoemission of the plasmonic field enhancement near the NP in terms of the averaged enhancement factor η eff R ( a , I 0 ) < η ( a , I 0 ) [64]. The effective enhancement factor η eff R is expected to be smaller than the plasmonic field enhancement η(a, I ₀) due to the decreasing strength of the nanoplasmonic field for increasing polar angles and PE-distances and due to averaging over the PE dynamics. Repulsive PE – PE and attractive PE – residual-charge Coulomb interactions are represented within a simplified central-field picture, respectively, as the contributions η e − e R ( a , I 0 ) > 0 and η res R ( a , I 0 ) > 0 to the effective Coulomb interaction factor

Our measured large cut-off energies indicate PE – PE interactions to dominate over PE –residual charge interactions for all intensities and sizes, such that η C R ( a , I 0 ) > 0 . For the laser parameters used in this study, the effective interaction time is determined at numerical convergence as ωt _f = 120.75 ≈ 2ωτ. Equation (3) and Table 1 explain qualitatively our experimental and simulated results in Figure 4 in terms of material characteristics and the NP size. The PE yield and cutoff energy tend to increase with decreasing work function, increasing plasmonic field enhancement, and increasing NP diameter [26], [49]. The leading factor U _p (I ₀) in Eq. (3) is linear in the laser peak intensity I ₀. We further note that the magnitude of the linear shift of the peak position in ATI spectra measured from sharp metal tips [70] was previously attributed to the plasmonic field enhancement of the incident IR pulse [71], in support of the evidence we find for strong plasmonic effects on photoemission from metal NPs.

Figure 4:

Simulated and experimental PE cutoff energies (a–d) and simulated yields (e–h) for gold, silver, and platinum nanospheres. (a,b,d,e,f,h) show PE cutoff energies and yields for gold and silver nanospheres with diameters of 10, 60, and 100 nm. (c,g) display PE cutoff energies and yields for gold and platinum nanospheres with a diameter of 70 nm. The gold – platinum comparisons is plotted against a gray background, for better distinction from the silver – gold results.

Figures 4a, b, and d present measured and simulated PE cutoff energies for gold and silver nanospheres. Figure 4c shows PE cutoff energies for platinum nanospheres with a diameter of 70 nm. We numerically simulated cutoff energies as the energy up to which 99.5 % of the net PE yield has accumulated [26], [49]. The experimental cutoff energies were extracted from the VMI maps as described in more detail in Refs. [39], [50].

Although experimental and simulated values are comparable in magnitude and show the same general trends in their size and intensity dependence, their overall agreement in Figure 4 is not perfect. We attribute the remaining discrepancies in part to the following uncertainties: With regard to the numerical simulation, a serious uncertainty derives from our implementation of approximate modified Fowler–Nordheim tunneling rates. While our simulated spectra are background-free and not adjusted for the experimental spectral detection efficiency, both background yields and detection uncertainties affect the measured data. The measured PE yields are subject to focal-volume averaging, such that for a given laser peak intensity, NPs exposed to a large range of intensities contribute to the PE yield. We partially offset this focal-volume effect by generating histograms of the number of PEs emitted per laser shot versus the number of laser shots. We then selected for our data analysis laser shots that yield the largest numbers of detected PEs, assuming they have illuminated NPs with intensities close to the laser peak intensity I ₀. PE counts associated with these preselected shots were added to compose the VMI maps shown in this work. From these maps the cutoff energies were defined where the PE yield becomes indistinguishable from the background [39], [50]. Other effects contributing to the measurement error in PE yields and energies are related to the (i) predominant emission of slow electrons overexposing the center of our MCP detector for the larger NPs and higher peak intensities used in the present work, (ii) NPs deviating from the ideal spherical shape and nominal size, and (iii) NP surfaces being coated with residual chemicals. Due to detection inefficiencies, the recorded PE yield is smaller than the actual yield. This difference is more significant for larger NPs and higher laser peak intensities.

Simulated PE yields for the NP materials and sizes in Figures 4a–d are shown in Figures 4e–h, respectively. For the laser parameters and targets in this study, comparing different materials of equal size shows that plasmonic field enhancements vary slightly for the considered metals (Table 1). PE yields and cutoff energies are larger for smaller work functions. In general, for high PE yield, the second term in Eq. (3) dominates, leading to larger cutoff energies. However, for the low PE yield in Figure 4(e), the second term in Eq. (3) becomes negligible and plasmonic enhancement dominates. Since the plasmonic field enhancement is slightly larger for gold than for silver, the cutoff energy is slightly higher for gold than for silver, especially at high intensities.

Figure 5 compares the simulated and experimental cutoff energies in Figure 4, scaled by the incident-laser ponderomotive energy U _p (I ₀). The error bars for the scaled cutoff energy are interpolated and shown as light- and dark-colored bands. They are calculated as

where ΔE _eV represents the uncertainty in the cutoff energy in units of eV. The light shaded regions in Figure 5 correspond to the error range given by the first term in Eq. (5), representing the uncertainty in the U _p –scaled cutoff energy. The darker shaded regions additionally account for a 10 % uncertainty in the laser peak intensity, as described by the second term in Eq. (5). This conservatively assumed accuracy of 10 % is of the order of the estimated intensity accuracy in our experiment (cf., Section 2.2) and clearly exceeds the typical calibration standard for laser peak intensities of 1.3 % [72]. The lighter regions exclude this intensity-related uncertainty.

Figure 5:

Simulated and experimental PE cutoff energies in Figures 4a–d in units of the incident-laser ponderomotive energy U _p (I ₀) ∼ I ₀.

In striking contrast to atomic targets, cutoff energies for NPs are significantly higher and exhibit a nonlinear dependence on the laser peak intensity I ₀. U _p –scaled cutoffs reach maxima that vary with the size and composition of the NPs. The emission saturation effect discussed in Section 3.1 occurs with increasing laser peak intensity I ₀. At low laser peak intensity and far from the above-introduced emission saturation intensity, increasing the laser peak intensity leads to a rapid increase in yield and, consequently, an increase in the cutoff energies in Figures 4 and 5. In contrast, at high laser peak intensity and close to the emission saturation intensity, a further increase in the intensity results in a slow yield increase and, consequently, an increase in the cutoff energy in units of eV in Figure 4 and decreasing U _p -scaled cutoffs in Figure 5.

For the considered transition metals and NPs of the same size, the simulated cutoff energies tend to converge to the same value at laser high intensity. The reason for this behavior is that – at high intensities – differences in the relevant material properties, i.e., different dielectric responses (resulting in material dependent plasmonic field enhancements) and different work functions (accounting for different net PE yields), become negligible, since these intensities are very close to the emission saturation intensity. In other words, at such high intensities, residual-charge and repulsive-Coulomb interactions between a large number of PEs become the primary elementary interactions that determine the shape of the PE spectrum at high PE energies and thus the cutoff energy.

A similar convergence to intensity-independent cutoff energies was observed for dielectric NPs [22]. The authors’ numerical simulations show a decreasing intensity and material dependence of the U _p –scaled cutoff energy due to dominant many-particle interactions, similar to our finding, albeit at larger laser intensities. The onset of this saturation effect occurring at larger intensities for dielectric, as compared to metal, NPs is expected, since higher field strengths are required for dielectric materials to release sufficiently many electrons for generating overwhelming PE Coulomb interactions.

In view of the PE spectra and cutoff energies discussed in this work being determined by complex strong-field driven many-electron interactions [26], the main characteristics in our measured spectra and cutoff energies and trends in their dependence on the NP size and laser-intensity are reasonably well reproduced by our numerical simulation, while some discrepancies remain. An important uncertainty in our simulation derives from our use of modified Fowler-Nordheim tunneling rates [26], as mentioned above. A more realistic account of electron tunneling at the NP surface, possibly based on density-functional theory, will likely be a key improvement for simulating more accurate PE yields. Expanding on the discussion of experimental errors given above in relation to the cutoff energies in Figure 4, we note that detector saturation decreases the reliability of the low-energy portion of our spectra. While this 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 measurements with the largest NPs and highest laser peak intensity presented in this work, due to larger numbers of emitted PE per laser shot [49]. Furthermore, the unequal PE yield for different NP sizes imposed a practical limit on the laser intensities. Small 10 nm NPs produce a much smaller PE yield than 100 nm NPs. We therefore increased the laser intensity for the small particles, as the signal for the lowest intensity was insufficient to generate reliable VMI spectra for the determination of cutoff energies. In contrast, for the larger particles, our larger intensities saturate the MCP, rendering the recorded VMI spectra useless for determining cutoff energies. These experimental constraints lead to the seemingly large range of, in part unequal, intensities we selected for different nanoparticle sizes.