Strong-field photoemission in nanotip near-fields: from quiver to sub-cycle electron dynamics

Abstract

Metallic nanotips exhibit large electric field enhancements over an extremely broad bandwidth spanning from the optical domain down to static fields. They therefore constitute ideal model systems for the investigation of the inherent frequency scalings of highly nonlinear and strong-field phenomena. Here, we present a comprehensive study of strong-field photoemission from individual metallic nanotips. Combining high local fields and variable-wavelength mid-infrared pulses, we investigate electron dynamics governed by the nanoscale confinement of the optical near-field. In particular, we characterize a transition to sub-cycle, field-driven electron acceleration. The experimental findings are corroborated by semiclassical calculations within a two-step model.

Appendices

Appendix 1: Additional Figures

See Figs. 7 and 8.

Fig. 7

Electron energy spectra for a large set of wavelengths and increasing laser intensity (from green to red). Incident field strengths were chosen to yield similar emission rates and range from 0.4 to $8.3\,{\text{V}}/{\text{nm}}$. The specific range of field strengths at each wavelength can be inferred from Fig. 8. A bias voltage of $-8\,{\text{V}}$ was applied to the nanotip. With increasing wavelength, the electrons are accelerated to higher energies, accompanied by a qualitative change of the spectral shape

Fig. 8

Incident field strength dependence of emitted charge per laser pulse for various wavelengths (double-logarithmic scale). The shape of the curves is basically wavelength-independent, as expected for adiabatic tunneling emission

Appendix 2: Semiclassical two-step model

In a one-dimensional description of strong-field photoemission, known as the Simpleman’s model [41], the electrons are regarded as classical point-particles that are accelerated in an oscillating electromagnetic field. The model was developed by Corkum [4] and others [37] for strong-field ionization of gases and is based on a separation of the process in two steps: Electron emission and subsequent acceleration of free electrons in the optical field. Here, it is applied to optical field emission from solid surfaces and further adapted to light-electron interaction in localized fields.

In the first step, the strong electric field bends the potential of the solid at the surface to form a tunneling barrier for the electrons. The emission process is approximated as static field emission, and the emission rate j is calculated via the Fowler–Nordheim equation [42, 43]

$$\begin{aligned} j = \frac{e^3F^2}{16\pi ^2\hbar t^2(y) \Phi }\cdot \exp \left( -\frac{4}{3}\frac{\sqrt{2m}v(y)\Phi ^{3/2}}{e\hbar F}\right) , \end{aligned}$$

(2)

wherein the electric field F is given by the time-dependent optical field strength within the adiabatic approximation. Here, $\hbar$ denotes the reduced Planck constant, $\Phi$ the work function of gold, and v(y) and $t^2(y)$ are correction terms [44, 45], accounting for the Schottky effect. In contrast to atomic gases, the spatial symmetry is broken by the metal surface. Emission only occurs for electric fields pointing toward the metal half space.

After tunnel emission, the metal potential is neglected (strong-field approximation) and the classical equation of motion is solved for a free electron in an oscillating electric field. The initial velocity is assumed to be zero. Depending on the time of emission, the electron might return to the metal surface. Here it can be reabsorbed, leading to the emission of a photon (high-harmonic generation). Alternatively, the electron is elastically or inelastically scattered. For this experiment, only the last two cases are relevant.

An exemplary simulation result is shown in Fig. 9. For each possible time of emission, the emission rate and final kinetic energy are calculated. The kinetic energy spectrum is then given by weighting the final kinetic energies by the emission rate.

Fig. 9

Exemplary simulation result (zoom into central cycle): In the first step, the time-dependent emission rate (gray) is calculated within the Fowler–Nordheim model (Eq. 2). Subsequently, the final kinetic energy of the electrons is determined from classical point-particle propagation (red and black line). Weighting the energies by the corresponding emission rate yields the kinetic energy spectrum. Simulation for $\delta = 1$, $l_F = 50\,{\text{nm}}$, $\lambda = 8\,\upmu {\text{m}}$, $\alpha = 15$ at a local field of $F=15.8\,{\text{V}}/{\text{nm}}$

The spatial inhomogeneity of the optical near-field strongly influences the electron acceleration. The consequential cutoff scalings, shown in Fig. 10, deviate from the common $2\,U_{\text {P}}$ or $10\,U_{\text {P}}$ scalings [40] which are observed in conventional far-field foci.

Fig. 10

Simpleman model calculations. a In a spatially homogeneous field ($\delta \rightarrow \infty$), the electron cutoff energy for direct electrons is given by $2\,U_{\text {P}}$ (black dashed line). Suppressed back-acceleration increases the cutoff energy in the intermediate regime. In the sub-cycle regime ($\delta \ll 1$), the electron leaves the enhanced near-field within an optical half-cycle, acquiring less kinetic energy. Experimentally, the $\delta$-parameter can be controlled by changing b wavelength or c intensity. Simulations were performed for field enhancement $\alpha =6$, pulse duration $\tau = 80\,{\text {fs}}$, local optical field strength $F=10\,{\text{V}}/{\text{nm}}$ (b) and $\lambda =8\,\upmu {\text{m}}$ (c). d Length scales for photoemission in diffraction limited laser foci (top) and at nanostructures (bottom). In the latter case, the electron’s quiver amplitude may exceed the spatial near-field extension

Appendix 3: Energy-dependent backscattering

Generally, the rescattering probability depends on the recollision energy of the scattering electrons at the surface. In the model of a one-dimensional potential step, higher energies result in a reduced reflectivity (Fig. 11d), neglecting lattice induced corrugation of the scattering potential. At longer wavelengths, the electrons are accelerated to higher impact energies (Fig. 11a, b); therefore, the contribution of rescattered electrons to the spectra decreases and the rescattering plateau washes out (cf. Fig. 11c). Taking into account the finite energy resolution of the TOF spectrometer,⁶ this may explain the experimentally observed shape of the energy spectra, where the high-energy part decays slower compared to simulations.

Fig. 11

Spectral signature of backscattered electrons. Emission phase dependence of electron final kinetic energy (red) and kinetic energy at the moment of return to the nanotip’s surface (shades of blue) in the quiver (a) and sub-cycle regime (b). The number of collision events increases with the emission phase (gray). For long wavelengths (b), backscattering is shifted to later emission phases and the kinetic energies are substantially larger. c) Spectral shapes for different rescattering models. Assuming a one-dimensional potential step of depth $V_0=-(E_F+\phi )$, the rescattering plateau is suppressed (blue) in comparison with constant rescattering probabilities (red, green). Simulation for $\lambda = 4\,\upmu {\text{m}}$ and $V_0=10.03\,{\text{eV}}\,{\text{nm}}$. d) Corresponding energy-independent (green) and energy-dependent (blue) reflectivities