We develop analytical models of optical-field-driven electron tunneling from the edge and surface of free-standing two-dimensional (2D) materials. We discover a universal scaling between the tunneling current density (J) and the electric field near the barrier (F): In(J/|F|β) ∝ 1/|F| with β values of 3/2 and 1 for edge emission and vertical surface emission, respectively. At ultrahigh values of F, the current density exhibits an unexpected high-field saturation effect due to the reduced dimensionality of the 2D material, which is absent in the traditional bulk material. Our calculation reveals the dc bias as an efficient method for modulating the optical-field tunneling subcycle emission characteristics. Importantly, our model is in excellent agreement with a recent experiment on graphene. Our results offer a useful framework for understanding optical-field tunneling emission from 2D materials, which are helpful for the development of optoelectronics and emerging petahertz vacuum nanoelectronics.

We revise the Fowler–Dubridge model for multiphoton over-barrier photoemission from two-dimensional (2D) materials to include the effects of reduced dimensionality, non-parabolic, and anisotropic energy dispersion of 2D materials. Two different directions of electron emission are studied, namely, vertical emission from the surface and lateral emission from the edge. Our analytical model reveals a universal temperature scaling of Tβ with β = 1 for the surface emission over a wide class of 2D materials and β = 3/2 for the edge emission from the 2D material with anisotropic parabolic energy dispersion, which are distinct from the traditional scaling of β = 2 originally derived for the traditional bulk materials. Our comparison shows good agreement with two experiments of photoelectron emitted from graphene for both surface and edge emission. Our calculations also show that the photoelectron emission is more pronounced than the coexisting thermionic emission for materials with low temperature and Fermi energy. This model provides helpful guidance in choosing proper combinations of light intensity, temperature range, and type of 2D materials for the design of photoemitters, photodetectors and other optoelectronics.

In this paper, we present an overview of the novel scaling laws of thermionic emission (TE), field emission (FE) and photoemission (PE) for two-dimensional (2D) materials. For these emerging materials, we express the emission models in the form of $\ln(J/K^{\beta})\propto 1/K$ , where J is the emission current density, $K$ is temperature (T) for TE and $K$ is dc or optically field for FE and PE. Here, the scaling is $\beta = 3/2$ and $\beta = 1$ for electron emission in lateral (or edge) and vertical (or surface) direction, respectively, which is different from the traditional scaling of $\beta = 2$ for bulk 3D materials.

We develop analytical models of optical-field electron tunneling from the edge and surface of two-dimensional (2D) materials, including the effects of reduced dimensionality, non-parabolic energy dispersion, band anisotropy, quasi-time dependent tunneling and emission dynamics indueced by the laser field. We discover a universal scaling between the tunneling current density $J$ and the laser electric field $F$: In($J/|F|^{\beta})\propto1/|F|$ with $\beta = 3 / 2$ in the edge emission and $\beta = 1$ in the vertical surface emission, which both are distinctive from the traditional Fowler-Nordheim (FN) model of $\beta = 2$. The current density exhibits an unexpected high-field saturation effect due to the reduced dimensionality of 2D materials, which is completely different from the space-charge saturation commonly observed in traditional bulk materials. Our results reveal the dc bias as an efficient method in modulating the optical-field tunneling sub-optical-cycle emission characteristics. Importantly, our model is in excellent agreement with a recent experiment on graphene. Our findings offer a theoretical foundation for the understanding of optical-field tunneling emission from the 2D material system, which is useful for the development of 2D-material based optoelectronics and vacuum nanoelectronics.

We construct an analytical formulation for nonlinear photoelectron emission in a dc-biased metallic nanovacuum gap triggered by a laser field, by exactly solving the one-dimensional time-dependent Schrodinger equation. We theoretically investigate the photoelectron energy spectra and emission current from left-and right-side surfaces of the asymmetric nanojunction with various dc biases, laser fields, and gap distances. The underlying photoemission mechanisms transitioning form multiphoton over-barrier emission to photon-assisted field tunneling, and the spatiotemporal dynamics of electron transport inside the gap are analyzed in detail. Our calculation shows applying a dc field could greatly reduce the interference oscillation in the transmission current in the nanogap, due to the shift of dominant emission away from the multiphoton over-barrier regime. Our results demonstrate that, besides the dc bias, varying the gap spacing could strongly influence the rectification on the photoelectron emission in a dc-biased metal-vacuum-metal gap. Our study provides useful guideline to the design of ultrafast nanogap-based signal rectification devices, such as photoelectron emitters and photodetectors, by choosing an optimal combination of dc bias, gap spacing, and material properties.

Photoemission driven by ultrafast optical fields enables spatiotemporal control of electron motion with extremely high precision. Here, we present a quantum model for ultrafast photoelectron emission from a dc-biased surface induced by laser pulses of arbitrary duration, ranging from subcycle to continuous wave, by solving the time-dependent Schr\"odinger equation exactly. The single formulation is valid from photon-driven electron emission in low intensity optical fields to field-driven emission in high intensity optical fields. We find the emitted charge per pulse oscillatorily increases with pulse repetition rate, due to varying coherent interaction of neighboring laser pulses. For a well-separated single pulse, our results recover the experimentally observed vanishing carrier-envelope phase sensitivity in the optical-field regime. We also find that applying a large dc field to the photoemitter is able to greatly enhance the photoemission current and in the meantime substantially shorten the current pulse.

Ultrafast laser-induced electron emission from nanostructures is fundamentally important to the development of coherent electron sources, ultrafast electron microscopy, and novel nano-vacuum devices [1] - [3] . For ultrashort pulsed laser induced photoemission, numerical simulations are typically implemented to study the emission property. Simplified Fowler-Nordheim based models are widely used to calculate the photoemission rate, but it works only in the strong optical field regime. To clearly reveal the underlying emission characteristics in different emission regimes under ultrashort pulsed condition, we construct an exact analytical solution for photoelectron emission from a dc-biased surface illuminated by laser pulses, by solving the time-dependent Schrödinger equation [4] - [7] . The model is valid for arbitrary pulse length from sub-cycle to CW excitation, and for arbitrary pulse repetition rate. The single formulation is valid from photon-driven electron emission in low intensity optical fields to field-driven emission in high intensity optical fields. Our calculations reveal the coherent interaction between neighboring laser pulses on the photoelectron emission, and well recover the experimentally observed sinusoidal carrier-envelope-phase modulation to energy spectra [8] and vanishing carrier-envelope-phase sensitivity of photoemission charge in the optical-field regime [9] . We also find adding a large dc field to the photoemitter is able to greatly enhance the photoemission current and significantly shorten the electron emission pulse.

We construct an exact analytical theory for ultrafast photoelectron emission from a dc-biased surface driven by laser pulses, by solving the time-dependent Schrödinger equation. The single formulation is valid from photon-driven electron emission in low intensity optical fields to field-driven emission in high intensity optical fields. Our results show the coherent interaction of neighboring laser pulses on the photoemission, and recover the experimentally measured vanishing carrier-envelope-phase sensitivity accompanied by a π phase shift in the optical-field regime. We also find adding a large dc field to the photoemitter is able to greatly enhance the photoemission current and shorten the electron emission pulse.

By exactly solving the one-dimensional time-dependent Schrödinger equation, we construct an analytical solution for nonlinear photoelectron emission in a nanoscale metal–vacuum–metal junction driven by a single-frequency laser field, where the impact of image and space charges is neglected. Based on the analytical formulation, we examine the photoelectron energy spectra and emission current under various laser fields and vacuum gap distances. Our calculation shows the transition from direct tunneling to multiphoton induced electron emission as gap distance increases. In the multiphoton regime, the photoemission current density oscillatorily varies with the gap distance, due to the interference of electron waves inside the gap. Our model reveals the energy redistribution of photoelectrons across the two interfaces between the gap and the metals. Additionally, we find that decreasing the gap distance (before entering the direct tunneling regime) tends to extend the multiphoton regime to higher laser intensity. This work provides clear insights into the underlying photoemission mechanisms and spatiotemporal electron dynamics of ultrafast electron transport in nanogaps and may guide the future design of advanced ultrafast nanodevices, such as photoelectron emitters, photodetectors, and quantum plasmonic nanoantennas.

In this study, we propose to coat metal nano-emitters with an atomically thick dielectric to further enhance the laser field near the nanotips and therefore the photoemission current. An optical field enhancement factor of up to 400 is demonstrated by the full-wave optical simulation, depending on the geometry of the nano-emitter and resonance wavelengths. A quantum photoemission model, which is constructed by solving the time-dependent Schrödinger equation, is utilized to investigate the photoemission processes. It is found that, over a wide range of laser fields, the emission current density from the coated photoemitter is enhanced by at least 2 orders of magnitude as compared to the bare emitter. The optical field emission regime can be reached at a much smaller field strength compared to the bare emitter.

Resonant optical field emission refers to photoelectron emission from nanostructures due to strong optical electric field enhanced by plasmonic resonances. 1 It is usually limited by its low emission current and low quantum efficiency. We propose to boost strong-field photoemission by coating the nanopyramid field emitter with an atomic-thick dielectric coating layer. This layer greatly increases the optical field enhancement near the emitter tip, in the meanwhile, creates a reduced double tunneling profile with increased electron emission probability. We performed numerical optical simulations, and use a quantum photoemission model to predict the performance of coated field emitter. We find that with 1 nm thick layer (refractive index of 1.5) coated at the gold nanotip, the electron emission current density can be enhanced by at least two orders of magnitude over a wide range of laser intensity, thanks to the increased local optical field enhancement factor from 35 to 200 (or intensity enhancement factor of 40000). The emission current can further increase with coating of even higher refractive index. The coated field emitter may be selectively excited with different plasmonic resonant modes by choosing the wavelength of laser source, such that the electron emission patterns and the emission current are tunable. This theoretical study paves the way of developing a new category of high performance field emitters operating at tunable optical frequencies.

Photoelectron spectroscopy is an important method to study the composition and electronic structures of solid surfaces [1], [2]. In particular, time-resolved photoemission spectroscopy can directly probe the dynamics of photoexcited electrons, enabling the measurement of lifetimes of surface electronic states, such as the image-potential states, via the control of the time delay between the pump and probe photons [3]–[6]. Here, we study the time-resolved photoelectron emission from a dc biased metal surface illuminated by two-color lasers, using a recent quantum mechanical model based on the exact solution of time-dependent Schrödinger equation [7]–[9]. Our calculations recover the experimentally measured photoemission current modulation profile (both magnitude and shape) and reveal the underlying photoemission processes for different dc fields. We find that the dynamics of n-photon excited states due to two-color lasers depends strongly on the applied dc bias. When the dc field is small, all the n-photon orders of the spectra are modulated in the same way as a function of the relative phase delay of the two-color lasers, in agreement with experiments [10]. When dc field increases, various n-photon excited states behave differently with respect to time delay. In the meantime, applying a large dc field to metal surface significantly increases the emission current, which can strongly facilitate the experimental detection of photoelectron emission.

We propose to use two lasers of the same frequency to achieve interference modulation of photoelectron emission. Using a quantum mechanical model, we study the time-resolved photoelectron energy spectra and emission current modulation under different laser and dc fields. We find that strong interference modulation of photoemission can be easily achieved with two lasers of the same frequency, due to, on one hand, the straightforward access to the single-frequency laser pairs in experiments and, on the other hand, the low threshold value of the ratio of the laser fields for large modulation depth even with a strong dc field. Our study demonstrates the capability of using interference modulation by single-frequency laser pairs for practical measurements of time-resolved photoelectron energy spectra.

By solving the time-dependent Schrödinger equation, we construct an exact analytical solution for nonlinear ultrafast electron emission from a dc-biased metal surface illuminated by two-color laser fields. Our results show a large dc bias can significantly increase the photoemission current, while maintaining a strong current modulation with respect to the phase delay of the two-color lasers. Application of our model to time-resolved photoelectron spectroscopy shows the dynamics of n-photon exited states depends strongly on the dc field.

By proposing an atomically thick dielectric coating on a metal nanoemitter, we theoretically show that the optical field tunneling of ultrafast-laser-induced photoemission can occur at an ultralow incident field strength of 0.03 V/nm. This coating strongly confines plasmonic fields and provides secondary field enhancement beyond the geometrical plasmon field enhancement effect, which can substantially reduce the barrier and enable more efficient photoemission. We numerically demonstrate that a 1 nm thick layer of SiO2 around a Au-nanopyramid will enhance the resonant photoemission current density by 2 orders of magnitude, where the transition from multiphoton absorption to optical field tunneling is accessed at an incident laser intensity at least 10 times lower than that of the bare nanoemitter. The effects of the coating properties such as refractive index, thickness, and geometrical settings are studied, and tunable photoemission is numerically demonstrated by using different ultrafast lasers. Our approach can also directly be extended to nonmetal emitters, to-for example-2D material coatings, and to plasmon-induced hot carrier generation.

Photoelectron emission from solids, as driven by high-intensity lasers, offers a platform for the coherent control of electron motion on ultrashort spatiotemporal scales. By solving the time-dependent Schrodinger equation, we present an exact analytical solution for the nonlinear ultrafast electron emission from a dc-biased metal surface illuminated by two-color laser fields. We systematically examine the combined effects of a dc electric field and two-color laser fields. In addition to the remarkable tunability of electron emission processes due to interference from two-color laser fields, we find that a strong dc electric field not only opens up more tunneling emission channels, but also introduces intense modulation to the emission current. We find surprising results that strong current modulation (with respect to the phase difference of the two-color lasers) persists (>70%), even with a large dc bias (i.e., ratio of the electric fields for dc : fundamental laser : second-harmonic laser congruent to 1 : 0.5 : 0.07). In the meantime, the average emission current level increases by about three orders of magnitude relative to the case of zero dc bias. Application of our model to time-resolved photoelectron spectroscopy is exemplified and shows that the dynamics of the n-photon excited states depends strongly on the applied dc field. Our study suggests a practical way to maintain a strong modulation to high current photoemission, by the addition of a large dc bias for two-color laser-induced electron emission.

This paper presents an analytical model for ultrafast electron emission from a metal surface illuminated by two-color continuous laser fields. The exact solution is valid for arbitrary fundamental and harmonic laser frequencies, laser intensities, phase difference between the two lasers, metal work function and Fermi level. We found two-color laser fields can strongly modulate the emission current up to 99%. Our results are in excellent agreement with recent experiments.

This paper presents field emission (FE) from a single carbon nanotube (CNT) fiber with varying anode- cathode (AK) gap distances. We found that the field enhancement factor depends strongly on the finite AK gap distance, due to the combination of geometrical effects and possible fiber morphology change. The slope drop in the Fowler-Northeim (FN) plot of the FE data in the high voltage is related to the electrical contact resistance between the CNT fiber and the substrate.

Electron emission from solids driven by two-color lasers provides great flexibility for the control of electron dynamics in ultrashort spatiotemporal scales due to the interference effect. Here, we construct an analytical model for the highly nonlinear photoelectron emission from a metal surface illuminated by two-color laser fields, by solving the time-dependent Schrodinger equation. The exact solution is valid for arbitrary harmonic orders, laser intensities, phase difference between two lasers, and metal work function and Fermi level. We find two-color lasers can strongly modulate both the electron energy spectra and the emission current up to 99%. Using the same input parameters, our theoretical prediction for the photoemission current modulation depth (93.9%) is almost identical to the experimental result (94%) in [M. Forster et al., Phys. Rev. Lett. 117, 217601 (2016)].

Ultrafast electron emission is important to free electron lasers (FELs), laser acceleration of relativistic electrons and electron sources. It is also important to the development of novel vacuum nanoelectronics [1]–[3]. Two-color laser pulses have been used to produce ultrafast electron emission from a metal nanotip, showing great control of the dynamics and distribution of electrons [4]. We present an analytical model for ultrafast electron emission from a metal-vacuum interface due to two-color laser fields, by solving the time-dependent Schrodinger equation [5]. Our exact solutions are valid for arbitrary fundamental and harmonic laser frequencies, laser intensities, phase difference between the two-color lasers, metal work function and Fermi level. The results are in a good agreement with the experiments [4]. Different electron emission mechanisms such as multiphoton absorption or emission, optical field emission and single-photon induced over-barrier emission are revealed in a single formulation.