PHOTOEMISSION FROM BIASED METAL SURFACES: QUANTUM EFFICIENCY, LASER HEATING, DIELECTRIC COATINGS, AND QUANTUM PATHWAYS INTERFERENCE By Yang Zhou A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Electrical Engineering – Doctor of Philosophy 2022 ABSTRACT Electron emission from metal surfaces due to the illumination of laser fields is of great interest due to its broad applications ranging from electron sources to quantum information processing to attosecond physics. It also offers fundamental insights into electron dynamics and electronic band structures of materials. This thesis analytically studies the effects of laser (wavelength or frequency, and field strength) and cathode surface conditions (with or without dielectric coatings) on photoemission from biased metal surfaces, by exactly solving one- dimensional (1D) time-dependent Schrödinger equation. Our study provides better understanding of photoemission dynamics, laser heating effects on photoemission, and two-color laser coherent control of photoemission with a dc bias, and it is useful for the design of more stable and efficient emitters. First, we study photoemission from a metal surface using an analytical quantum model. It is found that shorter wavelength lasers can induce more photoemission from electron initial energy levels further below the Fermi level and, therefore, yield larger quantum efficiency (QE). The dc field increases QE, but it is found to have a greater impact on lasers with wavelengths close to the threshold (i.e., the corresponding photon energy is the same as the cathode work function) than on shorter wavelength lasers. When the laser field increases, QE increases with the laser field strength in the longer laser wavelength range due to the increased contributions from multiphoton absorption processes. The quantum model is compared with classical three-step model, the Fowler–DuBridge model, and the Monte Carlo simulation based on the three-step model. Even though those models have very different settings and assumptions, it is found that the scaling of QE of our quantum model agrees well with other models for low intensity laser fields. Next, we study photoemission from metal surfaces with laser wavelengths from 200 to 1200 nm (i.e., ultraviolet to near infrared). It is found that QE can be increased nonlinearly by the non- equilibrium electron heating produced by intense sub-picosecond laser pulses. This increase of QE due to laser heating is the strongest near laser wavelengths where the cathode work function is an integer multiple of the corresponding laser photon energy. The quantum model, with laser heating effects included, reproduces previous experimental results, which further validates our quantum model and the importance of laser heating. We then present an exact analytical theory for field emission from dielectric coated cathode surfaces, by solving the 1D Schrödinger equation with a double-triangular potential barrier introduced by the coating. The condition under which the emission current density from the coated cathode can be larger than the uncoated case is identified. Our quantum model is also compared with a modified Fowler-Nordheim equation for a double barrier, showing qualitatively good agreement. We further extend the exact quantum model for electron emission from metal surfaces coated with an ultrathin dielectric to photoemission. It is also found that a flat metal surface with a dielectric coating can photoemit a larger current density than the uncoated case when the dielectric has smaller relative permittivity and larger electron affinity. Resonant peaks in the photoemission probability and emission current are observed as a function of dielectric thickness or electron affinity due to the quantum interference of electron waves inside the dielectric. Our model is compared with the effective single-barrier quantum model and modified Fowler– Nordheim equation, for both 1D flat cathodes and pyramid-shaped nanoemitters. While the three models show quantitatively good agreement in the optical field tunneling regime, the present model may be used to give a more accurate evaluation of photoemission from coated emitters in the multiphoton absorption regime. Finally, we analyze the quantum pathways interference in two-color coherent control of photoemission using exact analytical solutions of the time-dependent Schrödinger equation. The theory includes all possible quantum pathways and their interference terms. It is found that increasing the intensity ratio of the second harmonic (2ω) laser to fundamental (ω) laser results in less contribution from the ω pathway (absorption of ω photons only) and more contribution from multicolor pathway (simultaneous absorption of both ω and 2ω photons) and 2ω pathway (absorption of 2ω photons only), and therefore stronger pathways interference and increased visibility larger than 95%. Increasing bias voltages shifts the dominant emission to processes with lower-order photon absorption, which sequentially decreases the interference between the ω and the 2ω pathways, and between single-color and multicolor pathways, leading to two peaks in the visibility as a function of dc field. ACKNOWLEDGEMENTS It would not be possible to write this thesis without the kind help and firm support from all the people around me. Here, I would like to express my sincere gratitude to them. I would like to first thank my advisor Prof. Peng Zhang for his help, support, and guidance over the past four years. It would not be possible to have this thesis without his help and guidance. He has always been patient and careful with every detail. I can still remember when he pointed out a punctuation error in slides just at his first glance. He has always been there whenever I need help. His advice really helps for both my academic work and personal life. I really appreciate it. I would also like to thank all the committee members, Prof. Verboncoeur, Prof. Baryshev, Prof. Fan, and Prof. Ruan for their valuable time and helpful guidance. Professor Verboncoeur’s expertise in vast fields is always valuable to my research during my PhD study. He has provided me fruitful suggestions. Thanks to him for his question and comments at IVEC 2022 in Monterey, CA, which really eased my nervousness at that moment. Due to the supportive group members, Dr. Yangyang Fu, Dr. De-Qi Wen, Dr. Asif Iqbal, Dr. Janez Krek, Dr. Sneha Banerjee, Dr. Yi Luo, Faisal and Ayush, I really enjoy the past four years in MSU. Thanks to Dr. Luo for helping me with my research and for helping me settle down when I first came to MSU. I would like to thank all the friends and people I met in work and life for their help and support. Thanks to Xiangkai Feng for helping me all the time, from preparing TOEFL and GRE, to everyday life when I first came to the US. Thanks for their accompany when I was in depression and anxiety in 2020. Finally, I would like to thank my parents and sisters for their support and encouragement all the time, especially my mother, who is a kind, persistent, and hard-working woman. Their love and care have always motivated me to move forward. iv TABLE OF CONTENTS CHAPTER 1 INTRODUCTION .................................................................................................... 1 1.1 Background ................................................................................................................... 1 1.2 Fundamentals of Photoelectron Emission from Metals ................................................ 4 1.3 Motivation and Organization of the Thesis ................................................................... 6 CHAPTER 2 AN ANALYTICAL QUANTUM MODEL FOR PHOTOEMISSION FROM METAL SURFACES...................................................................................................................... 9 2.1 Introduction ....................................................................................................................... 9 2.2 Analytical Quantum Model ............................................................................................. 10 2.3 Results and Discussion .................................................................................................... 13 2.4 Comparison with Classical Models ................................................................................. 22 2.5 Concluding Remarks ....................................................................................................... 28 CHAPTER 3 EFFECTS OF LASER WAVELENGTH AND LASER HEATING ON PHOTOEMISSION FROM BIASED METAL SURFACES....................................................... 31 3.1 Introduction ..................................................................................................................... 31 3.2 Effects of Laser Wavelength on Photoemission .............................................................. 32 3.3 Laser Heating Effects ...................................................................................................... 38 3.4 Concluding Remarks ....................................................................................................... 44 CHAPTER 4 FIELD EMISSION FROM METAL SURFACES WITH NANOSCALE DIELECTRIC COATINGS .......................................................................................................... 46 4.1 Introduction ..................................................................................................................... 46 4.2 Analytical Model ............................................................................................................. 47 4.3 Results and Discussion .................................................................................................... 49 4.4 Comparison with Modified Double-barrier Fowler-Nordheim Equation ........................ 59 4.5 Concluding Remarks ....................................................................................................... 61 CHAPTER 5 PHOTOEMISSION FROM METAL SURFACES WITH NANOSCALE DIELECTRIC COATINGS AND ITS ENHANCEMENT BY PLASMONIC RESONANCE .. 63 5.1 Introduction ..................................................................................................................... 63 5.2 Effective Single-triangular-barrier Quantum Model ....................................................... 64 5.3 1D Exact Analytical Quantum Model ............................................................................. 70 5.4 Comparison of Models .................................................................................................... 87 5.5 Concluding Remarks ....................................................................................................... 92 CHAPTER 6 TWO-COLOR COHERENT CONTROL OF PHOTOEMISSION FROM METAL SURFACES .................................................................................................................................. 94 6.1 Introduction ..................................................................................................................... 94 6.2 Analytical Quantum Models for Two-color Laser Induced Photoemission from Metal Surfaces ................................................................................................................................. 95 6.3 Quantum Pathways Interference Model .......................................................................... 97 6.4 Concluding Remarks ..................................................................................................... 104 CHAPTER 7 SUMMARY AND SUGGESTED FUTURE WORKS ........................................ 106 7.1 Summary........................................................................................................................ 106 7.2 Suggested Future Works on the Improvement of the Models ....................................... 107 7.3 Suggested Future Works on the Applications of the Models ......................................... 109 v BIBLIOGRAPHY ....................................................................................................................... 111 vi CHAPTER 1 INTRODUCTION 1.1 Background Electron emission from metal surfaces due to the illumination of laser fields has received considerable attention due to its broad applications and has provided fundamental insights into electron dynamics and electronic band structures of materials [1]–[7]. The recent development of tailored ultra-short and intense waveform-controlled laser fields [8]–[10] and the advances in fabrication of nanostructures have enabled the characterization of photoemission processes on attosecond timescale and on nanometer spatial scale [7], [11]–[13], which makes the spatiotemporally controllable electron sources possible. Therefore, emerging applications, such as ultrafast electron microscopy [14]–[19], time-resolved scanning transmission electron microscopy [7], free electron lasers (FELs) [20]–[22], ultrafast electron diffraction [1], [6], [23], carrier-envelope detection [24], [25], and ultrafast light-wave driven nano-electronics [26]–[32], are accessible. The underlying processes of photoemission from metal surfaces have been investigated extensively. Multiphoton photoemission, above-threshold photoemission, photo-assisted field emission, and optical field emission are identified [33]–[39]. Laser-matter interaction is strongly affected by the frequency or wavelength of the oscillating light fields [40]. As wavelengths λ range from ultraviolet (UV, λ<380 nm) to terahertz (THz), different dominant emission processes and excited electron dynamics are observed. Single-photon photoemission has been observed in the broad UV wavelength range for various metallic cathodes [41]–[44]. With higher illumination intensities, electrons can be emitted by sub-work function energy photons (ℏ𝜔 < 𝑊, with 𝑊 work function of emitters). In this case, minimum n photons (𝑛 = 〈𝑊/ℏ𝜔 + 1〉, with 〈 〉 denoting the integer part) have to be absorbed simultaneously in order to excite an electron to escape the material. Nanostructured metallic emitters have enabled multiphoton photoemission, above-threshold photoemission, and optical field emission to occur at a relatively low incident laser intensity and high pulse repetition rates without damaging the emitter, due to significant enhancement of the laser electric field on the nanotip by the so-called lightning rod effect and surface plasmonic effects [36], [37], [45]–[47]. Hommelhoff et al. [45] experimentally demonstrated continuously tuning between the photofield emission (or photo-assisted field emission) and optical field emission from tungsten nanotips by varying the driving laser intensity with a wavelength centered at ~810 nm. Long-wavelength few-cycle laser pulses are a powerful 1 tool to control and manipulate electron dynamics from nanostructures on femtosecond and attosecond timescales and on an atomic spatial scale. It is common to find that the Keldysh parameter 𝛾 < 1 and therefore strong field regime can be reached at a relatively low laser intensity in long wavelength range, especially for emitters designed to be plasmon coupling with optical fields [37]. With laser wavelengths from 1 μm to 1.5 μm, Park et al. [48] observed the narrowing of the emission cone angle of the fastest electrons when laser intensity increases, which is ascribed to field-induced steering of sub-cycle electrons. Few-cycle midinfrared (up to 8 μm) laser pulses are applied to single plasmonic nanotips, and it is found that the electrons can escape the local field within a fraction of an optical half-cycle and electron quiver motion can be quenched [49]. Single-cycle terahertz pulses have also been demonstrated to have their capacity to control nanotip photoemission electron dynamics [50]–[52]. Incident lasers act not only as a photon source, but also as a heat source. Photoelectron emission processes are always accompanied by heating of the emitter surface to some extent due to high laser intensities. The heating effect pushes the Fermi tail of the electron distribution closer to the vacuum level. Therefore, thermally enhanced multiphoton photoemission, thermionic emission and photon-assisted thermionic emission can occur during the process [42], [53]–[57]. For shorter laser pulses (several picoseconds to a few hundreds of femtoseconds), the action of laser can also lead to nonequilibrium between electron and lattice systems, which was predicted theoretically and demonstrated experimentally [58]–[61]. For femtosecond lasers (<100 fs), the e-e collision is not fast enough to reach the internal thermalization during the laser pulse. Therefore, two-temperature model (TTM) is inadequate, and microscopic kinetic approach, e. g., Boltzmann’s equation [62], [63] and nonequilibrium Green’s-function theory [64], is required. Properties of the cathode materials also influence the fundamental mechanisms of photoemission. Metal cathodes, antimonide cathodes, and NEA (negative electron affinity) cathodes are commonly used as photocathodes for various applications, and each type has their own strengths and weaknesses [65], [66]. Metals are relatively robust, easy to prepare, and less sensitive to vacuum conditions. The metallic cathodes can be more attractive if they are designed with plasmon coupling to the optical field, which greatly enhances the local field at the metal- vacuum interface [67]. The emerging capabilities of tailoring composition and electronic structure of materials enable us to utilize heterostructure architectures, quantum wells, and 2 superlattices which consists of nanoscale structures, to design and fabricate photocathodes to satisfy the stringent requirements of high quantum efficiency, high brightness, and low emittance [68], [69]. Artificial coatings, such as, graphene, nano-diamond, silicon dioxide, and zinc dioxides, are proposed to be fabricated on cathodes to provide protection and improve quantum efficiency [70]–[75]. The work function of emitter is significantly altered by thin oxide films [76]. A reduction of >1 eV in the work function relative to Ag is observed for MgO monolayers on Ag [77]. Both theory and experiments have shown that the quantum efficiency can be improved with coatings [47], [70], [71], [75], [78], [79]. The cathode surface can be contaminated due to the bombardment by ions and electrons or can be oxidized under poor vacuum conditions. These modified surface conditions also influence the photoemission processes. In addition to novel nanostructured emitter tips, coherent lights consisting of a strong fundamental laser and a weak second harmonic, provide another essential tool to tune and control photoemission from nanotips. It was first applied in the study of electron dynamics of atoms and molecules [80]–[83]. The Hommelhoff team [84]–[88] reported their experiments on coherent control of photoemission from nanotips using two-color lasers. By varying the mixture ratio and sweeping the relative phase between those two fields, substantial modulation of photoemission current has been demonstrated, with a modulation depth or visibility of up to 97.5% [84]–[88]. In addition to the n-photon photoemission induced by respective lasers with 𝑛 = 〈𝑊/ℏ𝜔 + 1〉 for 𝜔-laser or 𝑛 = 〈𝑊/ℏ(2𝜔) + 1〉 for 2𝜔-laser, multicolor quantum pathways are also introduced, in which photons of different colors are simultaneously absorbed [85], [89], [90]. The strong modulation is attributed to the interferences among those pathways. By checking electron emission energy spectra, above-threshold photoemission from tungsten nanotip is observed [84], and the strong field regime is approached by applying strong few-cycle laser pulses with a nearfield intensity of ~4 TW/cm2 to a gold needle tip [88]. As another tuning knob, dc electric field suppresses high frequency terms, resulting in a reduced visibility [86]. Luo and Zhang developed an analytical quantum model by exactly solving the time-dependent Schrödinger equation, which includes multiphoton photoemission, above-threshold photoemission, photo-assisted field emission, and dc or optical field emission [91]–[93]. The model has shown good agreement with experiments and demonstrated the potential in measurement of time-resolved photoelectron energy spectra. 3 1.2 Fundamentals of Photoelectron Emission from Metals 1.2.1 Photoelectron Emission Mechanisms Figure 1.1 Mechanisms of photoemission from metal surfaces. (a) Multiphoton photoemission. An electron absorbs the minimum required number of photons to overcome the metal surface potential barrier to emit. (b) Above-threshold photoemission. An electron absorbs more than the minimum required number of photons to overcome potential barrier for photoemission. (c) Field emission. Direct tunneling of electrons through a narrowed potential barrier by dc field into vacuum. (d) Photo-assisted field emission. An electron is excited to a higher energy level by absorbing photons and tunnels through the narrowed potential barrier by dc field into vacuum. (e) Optical field emission. An electron tunnels through the periodically oscillating potential barrier induced by the strong laser field. A. Multiphoton absorption and Above-threshold photoemission In the multiphoton absorption process (see Fig. 1.1(a)), a single or several photons are absorbed, and the electron is elevated above the vacuum level. The minimum number n of photons required to overcome the potential barrier for electrons with initial energy at Fermi level is 〈𝑊/ℏ𝜔 + 1〉, with 〈 〉 denoting the integer part, 𝑊 being the work function of the emitter, ℏ being the reduced Planck constant, and 𝜔 being the angular frequency of the laser. The photoemission current in the multiphoton absorption regime follows a power scaling law [34], 𝐽 ∝ 𝐼 𝑛 or 𝐽 ∝ 𝐹12𝑛 , where I is the laser intensity and 𝐹1 is the strength of the laser field perpendicular to the emitter surface [94]. Absorption of photons (with an energy smaller than work function of the emitter) is considered to occur simultaneously, and thus multiphoton absorption always requires relatively intense lasers. The above-threshold photoemission requires absorption of more than the minimum required number of photons (see Fig. 1(b)), which is characterized by a plateau in photoelectron energy spectra [33], [38]. The broadening of the spectra also signifies the back-propagation and rescattering of electrons [95], which has also been observed in atoms [96]. B. Field emission and Photo-assisted field emission Field-induced emission exploits the quantum-mechanical tunneling process. Under a high 4 electrostatic field, the surface potential barrier of field emitters is bent down, electrons inside the emitter can tunnel through such a narrowed barrier into the vacuum, as shown in Fig. 1(c). Fowler and Nordheim first developed equations to predict the emission current by assuming the potential barrier under electric field as a triangular potential barrier [97]. The field emission current is found to depend on the dc field strength 𝐹 and work function of the emitter 𝑊. Fowler- Nordheim (FN) equation takes the following form [97]–[99], 3 𝐶1 (𝛽𝐹)2 𝐶2 𝑊 2 𝐽= exp [− ] (1.1) 𝑊 𝛽𝐹 where 𝐶1 = 1.54 × 10−6 A eV V −2 , 𝐶2 = 6.83 × 109 eV −3/2 V m−1 , 𝑊 (in eV) is the work function of the emitter, 𝛽 is the field enhancement factor, and 𝐹 (in V/m) is the electric field strength. Photo-assisted or laser-assisted electric field emission [45] is the process where electrons are first excited from their initial energy level to a higher energy level by absorbing photons, as shown in Fig. 1.1(d). The excited electrons see a much narrower potential barrier, and therefore, the tunneling probability is greatly improved. C. Optical field emission In the optical field emission or strong field tunneling regime (see Fig. 1(e)), the potential barrier is periodically narrowed down by the oscillating electric field of the laser such that electrons inside the metal can penetrate through the barrier in a fraction of optical half-cycle [49]. Therefore, a single electron beam with a duration of ~ half cycle of the incident laser can be generated, especially with the use of few-cycle, long-wavelength or low-frequency, and high- intensity laser pulses [50]. Optical field emission has been experimentally demonstrated for tungsten and gold nanotips [33], [45], [48], [100], [101], and tailored plasmonic nanostructures [36], [37], [46], [102]. D. Photo-assisted thermionic emission Due to the incident laser intensity, photon-electron interaction processes are always accompanied by heating of the surface to some extent. The heating effect puts the tail of the electron distribution closer to the vacuum level. Electrons in the tail of this distribution can be emitted into the vacuum by absorbing one or multiple photons or directly through thermionic emission [53], [56], [57]. Photo-assisted thermionic emission and laser-induced thermionic 5 emission have strong dependence on the intensity and length of the laser pulses [58]. For the relative low laser intensity and long laser pulse, single temperature heat conduction equation can describe the heating process [103]. For laser pulses of several hundreds of femtoseconds to a few picoseconds, laser radiation may lead to nonequilibrium between electron and lattice systems, which was predicted theoretically and demonstrated experimentally [59]. For femtosecond laser pulses, laser heating effects may be negligible [45]. 1.2.2 Keldysh Parameter The Keldysh parameter 𝛾 is an indicator of transition from multiphoton absorption to optical field tunneling in photoemission. 𝛾 is estimated as the ratio of two time scales [104], [105], 4𝜋𝑡𝑡 𝛾= , (1.2) 𝑇 where 𝑇 = 2𝜋/𝜔 is the cycle of the optical field, and 𝑡𝑡 = 𝑙/𝑣 is the time of an electron with velocity 𝑣 = √2𝑊/𝑚 tunneling through the potential barrier with a width 𝑙, 𝑊 is the effective work function of the metal and 𝑚 is the electron rest mass. Multiphoton photoemission is characterized by 𝛾 ≫ 1 , while optical field emission is characterized by 𝛾 ≪ 1. The exact transition point is difficult to determine, which is always accompanied by slope decrease in the plot of photoemission yield against laser intensity or field strength. The transition at 𝛾 ≈ 2 was confirmed by the reduction of the nonlinear power order in the electron yield as a function of the incident laser pulse energy [101]. More discussion for the transition region between these two extremes is given as 1 < 𝛾 < 𝑛0 with 𝑛0 = 〈𝑊/ℏ𝜔 + 1〉 [59]. 1.3 Motivation and Organization of the Thesis 1.3.1 Motivation of the Thesis As mentioned in Section 1.1, extensive studies on laser induced photoemission have been done over a broad range of wavelengths. Most studies use a single or several discrete laser wavelengths or a short range of laser wavelengths, which lacks a systematic analysis on the effects of laser wavelength and the accompanying laser heating effects on photoemission. Therefore, this thesis will first systematically investigate the photoemission current and quantum efficiency with a wide range of laser wavelength from 200 nm to 1200 nm, using a quantum model based on the exact solution of time-dependent Schrödinger equation, which is also 6 compared with classical models. Coatings on cathodes have strong influences on electron emission. However, there is lack of systematic study on the parametric scaling of electron emission from coated cathode surfaces and comprehensive understanding of the interplay of various parameters. Thus, the second part of this thesis will focus on developing analytical quantum theory for electron emission from metal surfaces with dielectric coatings and using the model to investigate the effects of dielectric coatings on electron emission. Two-color lasers and dc bias field can tune and control electron emission with great flexibility. Nevertheless, how the two-color lasers (fundamental and second harmonic laser fields, and their phase difference) and dc bias field influence the coherent control of photoemission is still not clearly understood. Therefore, the last part of this thesis will analyze the quantum pathway model using exact quantum theory for photoemission from biased metal surfaces under two-color lasers. 1.3.2 Organization of the Thesis In this thesis, we develop analytical quantum models to study photoemission from metal surfaces with and without nanoscale dielectric coatings under a dc bias, by exactly solving time- dependent Schrödinger equation (TDSE). The model includes photoemission, field emission, and thermionic emission with laser heating effects considered. We validate the quantum model for photoemission from bare metal surfaces by comparing it with classical models, and we study the effects of laser wavelength and corresponding laser heating effects on photoemission quantum efficiency, the effects of dielectric coatings on both field emission and photoemission, and two- color laser coherent control of photoemission processes. Chapter 2 presents a quantum model for photoemission from a metal surface due to the excitation of laser fields under a dc bias, by solving TDSE exactly. Quantum efficiency under various laser wavelengths (≤ 300 nm, in UV) and dc fields (≤ 0.1 V/nm) is studied. The model is compared with existing classical models, i.e., three-step model, Fowler-DuBridge model and three-step model based Monte Carlo simulation, for photoemission from both copper and gold metal surfaces. Chapter 3 presents the study of photoemission from a metal surface with laser wavelengths from 200 nm to 1200 nm (i.e., ultraviolet (UV) to near infrared (NIR)), using the quantum model developed in Chapter 2. The dominant electron emission mechanism, from different multiphoton 7 emission processes to dc or optical field emission, under different combinations of laser field strength (or laser intensity), wavelength, and dc bias field, is investigated. The parametric dependence of the quantum efficiency (QE) is analyzed in detail. Laser heating is taken into consideration by using the two-temperature model, and its effects on photoemission are examined. The quantum model, with laser heating effects included, is also compared with previous experimental results with excellent agreement. Chapter 4 presents an exact analytical theory for field emission from dielectric-coated cathode surfaces, by solving the 1D Schrödinger equation with a double-triangular potential barrier introduced by the coating. The effects of the cathode material (work function and Fermi energy), dielectric properties (dielectric constant, electron affinity and thickness), applied dc field strength, and cathode temperature are analyzed in detail. The quantum model is also compared with a modified double-barrier Fowler-Nordheim equation. Chapter 5 presents an analytical quantum model for photoemission from metal surfaces coated with an ultrathin dielectric, by solving the 1D TDSE subject to an oscillating double- triangular potential barrier due to the combined dc and laser fields. The effects of dielectric properties (thickness, relative permittivity, and electron affinity), laser field strength, and dc field on photoemission are systematically investigated. The model is compared with the effective single-barrier quantum model and modified Fowler-Nordheim equation, for both 1D flat cathodes and pyramid-shaped nanoemitters. Chapter 6 analyzes the quantum pathways interference in two-color coherent control of photoemission using exact analytical solutions of TDSE. The theory includes all possible quantum pathways and their interference terms. The effects of laser fields (the intensity mixture ratio of the second harmonic (2𝜔) to fundamental (𝜔) and their phase difference) and dc field on the weight of each pathway and the interference effects among them are analyzed in detail. Chapter 7 gives a summary and an outlook for future works. 8 CHAPTER 2 AN ANALYTICAL QUANTUM MODEL FOR PHOTOEMISSION FROM METAL SURFACES This chapter is based on the published journal paper “A quantum model for photoemission from metal surfaces and its comparison with the three-step model and Fowler–DuBridge model,” J. Appl. Phys., vol. 127, no. 16, p. 164903, Apr. 2020, doi: 10.1063/5.0004140, by Yang Zhou and Peng Zhang. 2.1 Introduction Laser-driven photoelectron emission is important to photoinjectors in free electron lasers and accelerators [69], [106], ultrafast electron microscopes [15], X-ray sources [107], femtosecond electron diffraction [108], and novel vacuum nanoelectronics [27], [28], [109]. The mechanisms of laser-driven photoemission from metal surfaces have been studied extensively both theoretically and experimentally [106], [110]–[117]. For decades, the fundamental models of photoemission remain those of classical treatment models, such as the three-step model [43], [44], [114]–[117], and the Fowler–DuBridge model [110]–[113]. There have also been recent interests in multi-photon absorption induced over-barrier emission for weak laser fields, photon- assisted tunneling and optical field tunneling emission for strong laser fields [38], [45], [62], [101], [118]. Quantum treatments of photoemission from metal surfaces assume the free electrons inside the metal are confined by a step potential barrier, which can be modulated by a dc field or a laser field [119]. Recently, an analytical model for electron emission from metal surfaces due to arbitrary combination of dc electric field and laser electric field was constructed by solving the time-dependent Schrödinger equation exactly [38]. The model has been extended to two-color laser induced electron emission [91], [92]. It includes the effects of lasers (wavelength and intensity), dc electric field, and metal properties (work function and Fermi level). However, it only considers electrons with initial energy at the Fermi level, without considering the contribution of photoemission from electrons with other initial energies inside the cathode. Here, we extend the quantum model for photoemission based on the exact solution of time- dependent Schrödinger equation, to include the effects of electron energy states distribution inside the metal, which is assumed to follow the Fermi–Dirac distribution. Electron emission mechanisms under various conditions are analyzed. The quantum efficiency (QE), defined as the number of emitted electrons per incident photon, is calculated from our quantum model. The 9 results are compared with the three-step model (both the Dowell’s analytical model [43], [44] and a simple Monte Carlo simulation), and the Fowler–DuBridge model. 2.2 Analytical Quantum Model Figure 2.1 Energy diagram for laser-driven photoelectron emission from a metal surface with a dc bias. The metal has Fermi level 𝐸𝐹 and original work function 𝑊0 . The dc field is 𝐹0 , and the laser electric field is 𝐹1 cos 𝜔𝑡, both of which are assumed perpendicular to the metal surface. The effective work function 𝑊 = 𝑊0 − 2√𝑒 3 𝐹0 /16𝜋𝜀0 with the Schottky effect. Left: Electrons inside the metal are assumed to follow the three-dimensional Fermi–Dirac distribution 𝑓(𝐸); center: The corresponding 1D electron supply function with longitudinal energy 𝜀 = 𝐸𝑥 is 𝑁(𝜀) (cf. Eq. (2.9) below); and right: the oscillating potential barrier due to laser and dc fields. Consider a 1D model [38], [91], [92] as shown in Fig. 2.1, in which it is assumed that electrons with initial longitudinal energy 𝜀 are emitted from the metal-vacuum interface at 𝑥 = 0, under the action of both a laser field 𝐹1 cos 𝜔𝑡, and a dc field 𝐹0 , where 𝐹1 is the magnitude of the laser field and 𝜔 is the angular frequency of the laser. The laser field is assumed to be perpendicular to the interface and cut off abruptly at the surface. This assumption may be justified by that the laser penetration depth is much smaller than the laser wavelength. Thus, the time-dependent surface potential barrier is [38], 0, 𝑥<0 Φ(𝑥, 𝑡) = { , (2.1) 𝐸𝐹 + 𝑊 − 𝑒𝐹0 𝑥 − 𝑒𝐹1 𝑥 cos 𝜔𝑡 , 𝑥 ≥ 0 where 𝐸𝐹 is the Fermi level of the metal, 𝑊 = 𝑊0 − 2√𝑒 3 𝐹0 /16𝜋𝜀0 is the effective work function including the potential barrier lowering by the Schottky effect due to the dc electric field 𝐹0 , 𝑊0 is the metal work function, e is the electron charge (positive), and 𝜀0 is the free space permittivity. Note that, though Eq. (2.1) assumes a linear potential in order to yield analytical solution, it has been verified in ref. [38] that this approach gives a very good 10 approximation of the more realistic nonlinear potential including image charge [120]. Due to the omission of the laser penetration inside the metal, the effects of electron-electron scattering and electron-photon scattering, which may happen when electrons inside the metal move to the surface, are also ignored in this model. The electron wave function 𝜓(𝑥, 𝑡) can be solved from the time-dependent Schrödinger equation, 𝜕𝜓(𝑥, 𝑡) ℏ2 𝜕 2 𝜓(𝑥, 𝑡) 𝑖ℏ =− + Φ(𝑥, 𝑡)𝜓(𝑥, 𝑡), (2.2) 𝜕𝑡 2𝑚 𝜕𝑥 2 where ℏ is the reduced Planck’s constant, m is the electron mass, and Φ(𝑥, 𝑡) is the laser field modulated surface potential barrier given in Eq. (2.1). For 𝑥 < 0, the solution to Eq. (2.2) is ∞ 𝜀 𝜀+𝑛ℏ𝜔 𝜓(𝑥, 𝑡) = 𝑒 −𝑖ℏ𝑡+𝑖𝑘0𝑥 + ∑ 𝑅𝑛 𝑒 −𝑖 ℏ 𝑡−𝑖𝑘𝑛 𝑥 , (2.3) 𝑛=−∞ where 𝜀 is the electron initial energy, 𝑅𝑛 is the reflection coefficient of the incident electron wave after absorption ( 𝑛 > 0 ) or emission ( 𝑛 < 0 ) of n photons. Equation (2.3) is the superposition of incident electron wave with the wave vector of 𝑘0 = √2𝑚𝜀 ⁄ℏ2 , and reflected electron waves from the metal-vacuum interface with the wave vector of 𝑘𝑛 = √2𝑚(𝜀 + 𝑛ℏ𝜔)⁄ℏ2 . Taking the Truscott transformation [121], the exact solution to Eq. (2.2) for 𝑥 ≥ 0 is found to be [38], [91], [92], ∞ 2𝑚𝐸 𝑖√ 2 𝑛 𝜉 ∑ 𝑇𝑛 𝑒 ℏ Θ(𝑥, 𝑡) , 𝐹0 = 0; 𝑛=−∞ 𝜓(𝑥, 𝑡) = ∞ (2.4) 𝑒 2 𝐹0 𝐹1 sin 𝜔𝑡 −𝑖 ∑ 𝑇𝑛 𝐺𝑛 (𝑥, 𝑡)Θ(𝑥, 𝑡)𝑒 ℏ𝑚𝜔 3 , 𝐹0 ≠ 0, { 𝑛=−∞ 𝑒𝐹 where 𝑇𝑛 is the electron wave transmission coefficient, 𝜉 = 𝑥 + (𝑚𝜔12) cos 𝜔𝑡 , 𝐸𝑛 = 𝜀 + 𝑛ℏ𝜔 − 𝜀+𝑛ℏ𝜔 𝑒𝐹1 sin 𝜔𝑡 𝑒 2 𝐹12 sin 2𝜔𝑡 𝐸𝐹 − 𝑊 − 𝑈𝑝 with 𝑈𝑝 = 𝑒 2 𝐹12 ⁄4𝑚𝜔2 , Θ(𝑥, 𝑡) = exp (−𝑖 𝑡+𝑖 𝑥+𝑖 ), ℏ ℏ𝜔 8ℏ𝑚𝜔 3 𝐺𝑛 (𝑥, 𝑡) = 𝐴𝑖(−𝜂𝑛 ) − 𝑖𝐵𝑖(−𝜂𝑛 ) with 𝜂𝑛 = (𝐸𝑛 /𝑒𝐹0 + 𝜉)(2𝑒𝑚𝐹0 /ℏ2 )1/3, and 𝐴𝑖 and 𝐵𝑖 are the Airy function of the first and second kind. Equation (2.4) is the transmitted electron wave traveling to the vacuum side, and it is the superposition of electron waves with the energy of 𝜀 + 𝑛ℏ𝜔 induced by different emission mechanisms, namely multiphoton absorption (n > 0), 11 tunneling (n = 0) and multiphoton emission (n < 0) [38], [91], [92]. Using boundary conditions that both 𝜓(𝑥, 𝑡) and 𝜕𝜓(𝑥, 𝑡)⁄𝜕𝑥 are continuous at x = 0, and taking the Fourier transform, we obtain, in the normalized form, ∞ 2√𝜀𝛿(𝑙) = ∑ 𝑇𝑛 (√𝜀 + 𝑙𝜔𝑃𝑛(𝑛−𝑙) + 𝑄𝑛(𝑛−𝑙) ), 𝑛=−∞ (2.5) 2𝜋 where 𝛿 is the Dirac delta function, 𝑃𝑛𝑙 = 1/2𝜋 × ∫0 𝑝𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) and 𝑄𝑛𝑙 = 2𝜋 1/2𝜋 × ∫0 𝑞𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) are the Fourier transform coefficients, with 2 ̅̅̅ 𝐹 2𝐹̅̅̅ 𝑖 1 3 sin 2𝜔𝑡 𝑖 21 √𝐸𝑛 cos 𝜔𝑡 𝐹̅1 𝑝𝑛 (𝜔𝑡) = 𝑒 4𝜔̅ 𝑒 ̅ 𝜔 , 𝑞𝑛 (𝜔𝑡) = 𝑝𝑛 (𝜔𝑡) [√̅̅̅ 𝐸𝑛 + sin 𝜔𝑡] , (2.6𝑎) 𝜔 for 𝐹0 = 0, and 𝐹̅1 sin 𝜔𝑡 1 𝑝𝑛 (𝜔𝑡) = 𝑧(𝜔 ̅𝑡̅)𝑠(𝛼𝑛 ), 𝑞𝑛 (𝜔𝑡) = 𝑧(𝜔 ̅𝑡̅) [ 𝑠(𝛼𝑛 ) + ̅̅̅ 𝐹0 3 𝑟(𝛼𝑛 )] , (2.6𝑏) 𝜔 ̅̅̅ 2 2𝐹 0 ̅̅̅ 𝐹1 sin 𝜔𝑡 ̅̅̅ 𝐹1 sin 2𝜔𝑡 for 𝐹0 ≠ 0 , with 𝑧(𝜔𝑡) = exp (−𝑖 +𝑖 3 ) , 𝑠(𝛼𝑛 ) = 𝐴𝑖(𝛼𝑛 ) − 𝑖𝐵𝑖(𝛼𝑛 ) , ̅3 𝜔 4𝜔 1 𝑟(𝛼𝑛 ) = 𝑖𝐴𝑖′(𝛼𝑛 ) + 𝐵𝑖′(𝛼𝑛 ), and 𝛼𝑛 = −[𝐸̅𝑛 /𝐹 ̅̅̅0 + (2𝐹̅1 /𝜔2 ) cos(𝜔𝑡)]𝐹 ̅̅̅0 3 . A prime denotes the derivative with respect to its argument. The quantities with a bar are in their normalized form, defined as 𝜀 = 𝜀 ⁄𝑊 , 𝜔 = ℏ𝜔⁄𝑊 , 𝑡 = 𝑡𝑊 ⁄ℏ , 𝑥 = 𝑥⁄𝜆0 , 𝜆0 = √ℏ2 /2𝑚𝑊 , 𝐹0 = 𝐹0 𝑒𝜆0 /𝑊 , 𝐹1 = 𝐹1 𝑒𝜆0 /𝑊 , 𝐸𝐹 = 𝐸𝐹 /𝑊 , 𝑈𝑝 = 𝑈𝑝 /𝑊 , and 𝐸𝑛 = 𝐸𝑛 /𝑊 = 𝜀 + 𝑛𝜔 − 𝐸𝐹 − 1 − 𝑈𝑝 . Therefore, the electron wave transmission coefficient 𝑇𝑛 can be obtained from Eq. (2.5). The probability of electron transmission can be calculated by 𝑤(𝜀, 𝑥, 𝑡) = 𝐽𝑡 (𝜀, 𝑥, 𝑡)/𝐽𝑖 (𝜀), where 𝐽(𝜀, 𝑥, 𝑡) = (𝑖ℏ/2𝑚)(𝜓∇𝜓 ∗ − 𝜓 ∗ ∇𝜓), 𝐽𝑖 and 𝐽𝑡 are the incident and transmitted electron probability current density respectively. The time-averaged probability of electron transmission from initial energy of ε after n-photon process, ⟨𝑤𝑛 (𝜀)⟩ , is, √𝐸𝑛 Im 𝑖 |𝑇𝑛 |2 , 𝐹0 = 0, √𝜀 ⟨𝑤𝑛 (𝜀)⟩ = (2.7) ( ) 1/3 𝐹0 |𝑇𝑛 |2 , 𝐹0 ≠ 0, { 𝜋√𝜀 and the total electron transmission probability 𝐷(𝜀) from initial energy 𝜀 is the sum of ⟨𝑤𝑛 (𝜀)⟩, 12 ∞ 𝐷(𝜀) = ∑ ⟨𝑤𝑛 (𝜀)⟩ . (2.8) 𝑛=−∞ The emission current density can thereby be obtained by, ∞ 𝐽 = 𝑒 ∫ 𝐷(𝜀)𝑁(𝜀)𝑑𝜀 , (2.9) 0 𝐸𝐹 −𝜀 𝑚𝑘𝐵 𝑇 where 𝐷(𝜀) is given in Eq. (2.8), and 𝑁(𝜀) = 2𝜋2ℏ3 ln (1 + 𝑒 𝑘𝐵 𝑇 ) is the supply function, with 𝑁(𝜀)𝑑𝜀 being the flux of electrons inside the metal impinging on the metal surface with longitudinal energy between 𝜀 and 𝜀 + 𝑑𝜀 across a unit area per unit time, calculated from free- electron theory of metal [122]–[124]. The quantum efficiency (QE) is defined as the ratio of the number of emitted electrons to that of incident photons, 𝐽/𝑒 𝑄𝐸 = , (2.10) 𝐼/ℏ𝜔 where 𝐼 is the intensity of the incident laser, which is related to the laser electric field as I [W/cm2] = 𝜀0 𝑐𝐹12 /2 = 1.33 × 1011 × (𝐹1 [V/nm])2 for linearly polarized plane waves. 2.3 Results and Discussion Figure 2.2 Time-averaged electron transmission probability ⟨𝑤𝑛 (𝜀)⟩ through nth channel (or n- photon process) from electron initial energy 𝜀 = 𝐸𝐹 , under various combinations of laser wavelengths 𝜆 and laser fields 𝐹1 , with dc field 𝐹0 = 0. The metal is assumed to be copper, with 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV. The laser intensity corresponding to laser field 𝐹1 (V/nm) is I [W/cm2] = 𝜀0 𝑐𝐹12 /2 = 1.33 × 1011 × (𝐹1 [V/nm])2 . Figure 2.2 shows the time-averaged electron transmission probability ⟨𝑤𝑛 (𝜀)⟩ through the n-photon process from copper, under various combinations of laser fields and wavelengths. The 13 laser wavelengths for Figs. 2.2(a) – 2.2(d) are 180 nm, 220 nm, 260 nm and 280 nm, corresponding to the photon energy of 6.89 eV, 5.64 eV, 4.77 eV and 4.43 eV, respectively. The applied dc field is 0. The laser fields 𝐹1 for lines in colors of purple, blue, green and orange, are 10 V/nm, 1 V/nm, 0.1 V/nm and 0.01 V/nm, corresponding to the laser intensity of 1.33 × 1013 , 1.33 × 1011 , 1.33 × 109 and 1.33 × 107 W/cm2 , respectively. The electron initial energy 𝜀 is assumed to be at the Fermi level 𝐸𝐹 . It is clear that the electron transmission probability through the nth channel increases when the laser field increases. The dominant emission process is through the single photon absorption induced over-barrier emission (𝑛 = 1). Calculations with applied dc fields 𝐹0 of up to 0.1 V/nm, show 𝐹0 has little effects on the transmission probability spectrum, indicating that the Schottky effect is negligible for such a small dc bias in the laser wavelength range of 180 nm – 280 nm on copper cathodes. Figure 2.3 The electron transmission probability 𝐷(𝜀) at initial energy of 𝐸𝐹 as a function of laser electric field 𝐹1 , for various laser wavelengths 𝜆 and dc fields 𝐹0 . The metal is assumed to be copper, with 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV. The effects of laser fields 𝐹1 (or laser intensity 𝐼), dc fields 𝐹0 , and laser wavelengths 𝜆 on the total electron transmission probability 𝐷(𝜀) from initial energy of 𝐸𝐹 is shown in Fig. 2.3. For the relatively small applied dc field (up to 0.1 V/nm), the electron transmission probability is well scaled as 𝐷(𝜀) ∝ 𝐹12 for wavelengths shown in Fig. 2.3, indicating the dominant single- photon process. This is consistent with the characterization based on the Keldysh parameter 𝛾 = 14 √𝑊/2𝑈𝑝 = 𝜔√2𝑚𝑊/𝑒𝐹1 , which is used to characterize the transition from multiphoton absorption (𝛾 ≫ 1) to strong field emission (𝛾 ≪ 1). The smallest Keldysh parameter is 𝛾 = 4.04 when 𝐹1 = 12 V/nm, 𝐹0 = 0.1 V/nm at 𝜆 = 260 nm, implying that strong-field emission can be neglected. When the laser wavelength increases from 180 nm to 260 nm, the corresponding photon energy decreases from 6.89 eV to 4.77 eV and becomes closer to the potential barrier seen by the electrons (i.e., 𝑊 ≅ 4.31 eV), leading to an increasing transmission probability 𝐷(𝜀). This is consistent with our previous study that the maximum transmission occurs when the photon energy equals to the potential barrier, ℏ𝜔/𝑊 ≈ 1 (cf. Fig. 6c of [38]). When the dc field increases, the potential barrier becomes narrower (Fig. 2.1), thus increasing the electron tunneling probability. In the meantime, the effective work function decreases because of the Schottky effect. The effects of dc field are more pronounced for the larger laser wavelength (or smaller photon energy) of 𝜆 = 260 nm, as shown in the inset of Fig. 2.3. When the dc field is strong, e.g., 𝐹0 = 5 V/nm, the dominant emission process becomes dc field emission (𝑛 = 0) when laser field is small, and electron transmission probability 𝐷(𝜀 = 𝐸𝐹 ) is independent of the laser wavelengths (not shown). The electron transmission probability 𝐷(𝜀) at different initial electron energy 𝜀 is shown in Fig. 2.4. The solid curves, which give 𝐷(𝜀) as a function of 𝜀 at various laser fields, show a stair- like behavior. Each “stair” indicates a specific n-photon absorption for electrons with initial energy in that range. The energy between step points corresponds to the laser photon energy ℏ𝜔. Take Figs. 2.4a(ii), (v) and (viii) for 𝜆 = 220 nm as an example, whose projections in the 𝐷 − 𝜀 plane and in the 𝐷 − 𝐹1 plane for a few selected curves are shown in Figs. 2.4b and 2.4c respectively. When 𝐹0 = 0, the first step point is at 𝜀 = 5.67 eV (Fig. 2.4b), which equals 𝐸𝐹 + 𝑊0 − ℏ𝜔, representing the electron initial energy threshold for one-photon emission. The second step point is near the bottom of the energy state. The difference between the two step points is 5.64 eV, which is the photon energy of the 220 nm laser. Electrons with initial energy in this range emit through two-photon absorption. The transmission probability 𝐷(𝜀) of electrons with initial energy 𝜀 in the same stair range keeps almost constant under the same laser field 𝐹1 , only with a slight decrease as 𝜀 increases in that stair range. The difference in 𝐷(𝜀) between stairs decreases when 𝐹1 increases. For a given 𝐹1 , when the applied dc field 𝐹0 increases, the step point shifts to smaller 𝜀, because the surface potential barrier (or effective work function) is lowered by the applied dc field. The transition between stairs becomes gradual when there is dc 15 field instead of a sharp transition when there is no dc field. Figure 2.4c shows 𝐷(𝜀) as a function of laser field 𝐹1 with various electron initial energies when the laser wavelength is 220 nm. When the electron is with initial energy of 2 eV and 4 eV, 𝐷(𝜀) is well scaled as 𝐷(𝜀) ∝ 𝐹12𝑛 = 𝐹14 for the applied dc field up to 0.1 V/nm, indicating the dominant two-photon absorption process. When the electron is with initial energy of 5 eV, the dominant emission is through two- photon absorption for dc field smaller than 0.05 V/nm. When the applied dc field 𝐹0 = 0.1 V/nm, which induces an effective work function of 𝑊 = 3.93 eV, the effective potential barrier seen by an electron at 𝜀 = 5 eV becomes 𝐸𝐹 + 𝑊 − 𝜀 =5.93 eV (compare to photon energy of 5.64 eV for 220 nm laser). The electron is emitted through single photon assisted tunneling (𝑛 = 1) in the laser field range from 𝐹1 = 0.001 V/nm to 1 V/nm. The dominant emission process shifts to high order channels (𝑛 = 2) when the laser field gets larger in the range from 𝐹1 = 1 to 10 V/nm, as shown in Fig. 2.4(c), which is consistent with our previous observation (cf. Fig. 2 of [38]). 16 Figure 2.4 Electron transmission probability 𝐷(𝜀) as a function of electron initial energy 𝜀, for different laser field 𝐹1 , dc field 𝐹0 , and laser wavelength 𝜆. The metal is assumed to be copper, with 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV. 17 Figure 2.5 Electron emission current density per electron initial energy 𝐽(𝜀) = 𝑒𝐷(𝜀)𝑁(𝜀) as a function of electron initial energy 𝜀 under various combinations of laser fields, dc fields and wavelengths. The metal is assumed to be copper, with 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV. The temperature is assumed at 𝑇 = 300 K. Erratum: in ref. [125], y-axis label should be 𝐽(𝜀)[A/ cm2 /𝐽] as for the scale in Fig. 5; here it has be converted to 𝐽(𝜀)[A/cm2 /𝑒𝑉]. Figure 2.5 presents the electron emission current density per energy 𝐽(𝜀) = 𝑒𝑁(𝜀)𝐷(𝜀) as a function of electron initial energy 𝜀 under various combinations of laser fields, dc fields and laser wavelengths. Most electrons are emitted from electron initial energy of a few eVs below the Fermi level, through single photon absorption. The range of the electron initial energy of single photon emission is determined by the incident laser photon energy. The shorter the laser wavelength, the larger the range, therefore, more electrons are emitted. Few electrons above the Fermi level are emitted, because there are fewer electrons distributed in that energy range at room temperature. The dc field extends the dominant electron emission range to a lower value, and therefore more electrons with smaller initial energy are able to emit. The electron emission 18 current density per energy 𝑒𝑁(𝜀)𝐷(𝜀) also shows stair-like dependence on the electron initial energy 𝜀, indicating different n-photon absorption process for electrons with initial energy in different “stairs”. The step points correspond to the initial energy threshold for an electron to emit after absorption of integer number of photons 𝐸𝐹 + 𝑊 − 𝑛ℏ𝜔. When the laser field 𝐹1 increases, the electron emission current density also increases, but the difference of emission current density between stairs gets smaller, especially for longer laser wavelength, which means multi-photon absorption process contributes more significantly to the electron emission. When the laser field 𝐹1 is small, as in Figs. 2.5(a)–2.5(c), the “stairs” are flat. When 𝐹1 increases, 𝑒𝑁(𝜀)𝐷(𝜀) from larger initial energy becomes smaller than that from smaller initial energy in the same “stair”, which is similar to the transmission probability 𝐷(𝜀) observed in Fig. 2.4. Figure 2.6 Total electron emission current density as a function of laser wavelength under various laser fields 𝐹1 and dc fields 𝐹0 . The metal is assumed to be copper, with 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV. The electron temperature is assumed at 𝑇 = 300 K. Figure 2.6 shows the total electron emission current density 𝐽 as a function of laser wavelength under various laser fields, as calculated from Eq. (2.9). It is clear that when laser field increases, the total emission current density also increases, especially in the longer wavelength. When the laser field is small (𝐹1 < 0.1 V/nm), there is an obvious drop in the current density at a longer wavelength. When the laser field is larger, the electron emission current density becomes less sensitive to the laser wavelength. The application of dc field increases the total electron emission current density, due to both narrowing and lowering of the 19 potential barrier. The increase of the emission current density due to the dc field is more significant for longer laser wavelengths. Figure 2.7 Total electron emission current density as a function of laser wavelength with different temperatures. The laser field 𝐹1 = 1 V/nm, and dc field 𝐹0 = 0 for (a) and 𝐹0 = 0.1 V/nm for (b). The inset in (b) is the supply function 𝑁(𝜀) under various temperatures. The metal is assumed to be copper, with 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV. Regarding the high power of the laser used for photoemission, the temperature near the surface of the metal could increase substantially. The effects of temperature on the emission current are shown in Fig. 2.7. The laser field 𝐹1 is 1 V/nm, and the temperature in the Fermi- Dirac function is set to be 𝑇 = 300, 500, 1000 and 2000 K. It is clear that higher temperature can increase the emission current density in the longer laser wavelength range. As the electron temperature increases, there are more electrons near or above the Fermi level which are able to emit and induce a larger emission current density. Electrons with initial energy above the Fermi level can emit through single photon absorption in the longer laser wavelength range, whose transmission probability is several orders higher than electrons with lower energy (cf. Fig. 2.4). Therefore, the electron emission current density is greatly enhanced in the longer laser wavelength range. In the shorter laser wavelength range, electrons mostly emit through single photon absorption over a wider range of initial energy around the Fermi level, therefore, the emission current density is insensitive to temperature. Figure 2.8 shows quantum efficiency (QE) as a function of laser wavelength and laser field for different dc fields, as obtained from Eq. (2.10). For smaller laser wavelength in the range of 20 180 nm – 260 nm, QE changes little when the laser field 𝐹1 increases from 0 to 10 V/nm, for a fixed dc field 𝐹0 . In this wavelength range, the electron emission is dominated by single photon absorption, giving the scaling of the emission electron current density as 𝐽 ∝ 𝐹12 , or 𝐽 ∝ 𝐼, thus, a constant QE ∝ 𝐽/𝐼 independent of 𝐹1 , as seen from Figs. 2.8(c) and 2.8(f). For laser wavelength in 260 nm – 300 nm, QE increases when the input laser field 𝐹1 increases. This is due to the increase in electron emission through multiphoton absorption (𝑛 > 1), as shown in Fig. 2.2. For the case of 300 nm, QE is increased by at least four orders of magnitude when 𝐹1 increases from 0.001 V/nm to 10 V/nm (cf. Fig. 2.8c with 𝐹0 = 0), indicating that multiphoton absorption contributes significantly in this case. Applying a dc field will increase QE, as shown in Figs. 2.8d – 2.8f. The increase of QE due to the dc field is more profound in the longer laser wavelength range for small laser field (e.g., 𝜆 in the range of 260 nm – 300 nm with 𝐹1 < 1 V/nm). It is also found that QE becomes less sensitive to the laser field 𝐹1 when dc field 𝐹0 is increased, especially for longer laser wavelength in 260 nm – 300 nm. 21 Figure 2.8 Quantum efficiency calculated from the quantum model under various laser fields in the laser wavelength range of 180 – 300 nm for dc field 𝐹0 of (a) – (c) 0 V/nm, and (d) – (f) 0.1 V/nm. The metal is assumed to be copper, with 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV. The temperature is assumed at 𝑇 = 300 K. 2.4 Comparison with Classical Models In this section, we compare our quantum model with the widely used three-step model (TSM) [43], [44], [114]–[117] and Fowler-DuBridge (FD) model [110]–[113]. A brief summary of those two classical models is provided below. 2.4.1 Three-step Model Photoemission from metal surfaces is described in three sequential independent steps. First, electrons are excited to states of higher energies after absorption of photons. Next, the excited electrons move to the surface, which involves physical processes such as electron-electron scattering in metals. Finally, electrons overcome the surface potential barrier and escape into the vacuum, with a probability depending on their momentum and the surface potential. The quantum efficiency of photoemission from metal surfaces, defined as the ratio of the number of emitted electrons to that of the incident photons, is the product of the probability of those three 22 steps [43], [44], 𝑄𝐸 = [1 − 𝑅(𝜔)] × ∞ 1 2𝜋 ∫𝐸𝐹+𝑊−ℏ𝜔 𝑑𝐸[1 − 𝑓𝐹𝐷 (𝐸 + ℏ𝜔)]𝑓𝐹𝐷 (𝐸) ∫cos 𝜃𝑚𝑎𝑥 (𝐸) 𝑑(cos 𝜃) 𝐹𝑒−𝑒 (𝐸, 𝜔, 𝜃) ∫0 𝑑𝜙 ∞ 1 2𝜋 , (2.11) ∫𝐸𝐹−ℏ𝜔 𝑑𝐸[1 − 𝑓𝐹𝐷 (𝐸 + ℏ𝜔)]𝑓𝐹𝐷 (𝐸) ∫−1 𝑑(cos 𝜃) ∫0 𝑑𝜙 where 𝑅(𝜔) is the reflectivity at frequency of ω, the laser is assumed incident perpendicular to the metal surface; 𝑓𝐹𝐷 (𝐸) = 1/(1 + exp[(𝐸 − 𝐸𝐹 )/𝑘𝐵 𝑇]) is the Fermi-Dirac function, describing the electron energy states distribution inside the metal; W is the effective work function, including the Schottky effect, which is the maximum of the Schottky potential composed of both image charge field and applied dc field F0; 𝐹𝑒−𝑒 (𝐸, 𝜔, 𝜃) is the probability that electrons survive the e-e scatterings to reach the surface; 𝜃 is the angle between the velocity of the electron and the surface normal, and 𝜙 is the azimuthal angle on the surface. cos[𝜃𝑚𝑎𝑥 (𝐸)] = √(𝐸𝐹 + 𝑊)/(𝐸𝐹 + ℏ𝜔) gives the maximum angle of the electron velocity with respect to the normal of the surface, at which an electron can reach the surface and may eventually escape. There are assumptions made to simplify the model. First, the metal is assumed to be at low temperature. Therefore, the Fermi-Dirac function can be approximated as a step function, with fully occupied states below EF and empty states above EF. Second, the photon energy of input laser is assumed to be near the threshold of photoemission, so that most of the emitted electrons are with velocity normal to the surface, and therefore the 𝜃 dependence of 𝐹𝑒−𝑒 (𝐸, 𝜔, 𝜃) can be ignored. Implied in Eq. (2.11) is that electrons with the momentum normal to the surface greater than the critical barrier momentum may escape, which satisfies 𝑝⊥2 /2𝑚 > 𝐸𝐹 + 𝑊, where 𝑝⊥ is the momentum normal to the surface. Thus, the model in Eq. (2.11) can be simplified as [43], [44] (𝐸𝐹 + ℏ𝜔) 𝐸𝐹 + 𝑊 𝐸𝐹 + 𝑊 𝑄𝐸 = [1 − 𝑅(𝜔)]𝐹𝑒−𝑒 [1 + − 2√ ], (2.12) 2ℏ𝜔 𝐸𝐹 + ℏ𝜔 𝐸 + ℏ𝜔 𝜆𝑜𝑝𝑡 (𝜔) where 𝐹𝑒−𝑒 (𝜔) = 1/ (1 + 𝜆 ) , 𝜆𝑜𝑝𝑡 (𝜔) = 𝜆/4𝜋𝑘(𝜔) is the optical penetration depth, 𝜆 is 𝑒−𝑒 (𝜔) the laser wavelength and 𝑘(𝜔) is the imaginary part of the refractive index, and 𝜆𝑒−𝑒 (𝜔) = 3/2 2𝜆𝑚 𝐸𝑚 1 is the mean free path of electron-electron scattering, with 𝜆𝑚 being a known ℏ𝜔√𝑊 1+√𝑊/ℏ𝜔 value of electron-electron scattering mean free path for electrons at energy 𝐸𝑚 above the Fermi level. Note that Eq. (2.12) is only a limiting case of TSM and is valid when the photon energy 23 ℏ𝜔 is close to the work function 𝑊 . When ℏ𝜔 is not close to 𝑊 , scattered electrons with transverse energies and multiphoton processes may contribute greatly to the total emission and have to be considered in a more general three-step formalism [126]. 2.4.2 Fowler-DuBridge Model Fowler-DuBridge model for photoemission [110]–[113] are based on the following assumptions: (a) electrons inside a metal follow Fermi-Dirac distribution, and electrons are distributed uniformly in the momentum space; (b) the probability of a photon absorbed by an electron is independent of the electron’s initial energy state; (c) only the electron momentum component normal to the surface is increased by the absorption of photon; (d) an electron can escape from the surface, if the electron momentum normal to the surface is greater than the threshold determined by the metal work function; (e) the quantum efficiency is proportional to the number of electrons impinging on the surface, per unit area per unit time, whose kinetic energy associated with the momentum normal to the metal surface is greater than the work function of the metal. For the laser wavelength shorter than the threshold wavelength, the single photon emission is dominant and multiphoton processes are ignored, which yields [110], ∞ ∞ ∞ 2 𝑝𝑧 /𝑚 𝑄𝐸 = 𝑎1 (1 − 𝑅) 3 ∫ 𝑑𝑝𝑥 ∫ 𝑑𝑝𝑦 ∫ 𝑑𝑝𝑧 , (2.13) ℎ 𝑝𝑥 2 + 𝑝𝑦 2 + 𝑝𝑧 2 𝐸𝐹 −∞ −∞ 𝑝1 exp ( − )+1 2𝑚𝑘𝐵 𝑇 𝑘𝐵 𝑇 where px and py are the momentum along the metal surface, pz is the momentum normal to the surface, 𝑝1 = √2𝑚(𝑊 + 𝐸𝐹 − ℏ𝜔) is the minimum momentum normal to the surface required for an electron to overcome the potential barrier to emit, ℎ is the Planck’s constant, kB is the Boltzmann constant, T is the electronic temperature, 𝐸𝐹 is the Fermi energy, R is the metal surface reflectivity, 𝑅 = [(𝑛0 − 1)2 + 𝑘 2 ]/[(𝑛0 + 1)2 + 𝑘 2 ] when the laser is incident normal to the surface, 𝑛0 is the real part of the refractive index, 𝑘 is extinction coefficient or the imaginary part of the refractive index, α1 is a constant, which can be experimentally determined, a1= 5×10- 18 m2/A [127] for copper, and a1= 7×10-18 m2/A for gold, which is obtained by fitting with TSM results. By performing the integral in Eq. (2.13), the quantum efficiency can be expressed as [110]– [113] ℏ𝜔 − 𝑊 𝑄𝐸 = 𝑎1 (1 − 𝑅)𝐴𝑇 2 𝐹 ( ), (2.14) 𝑘𝐵 𝑇 24 where A=120 A/cm2/K2 is Richardson’s constant, F(x) is Fowler’s function, which is [110]–[112] e2𝑥 e3𝑥 e𝑥 − + 2 − ⋯, 𝑥 < 0, 22 3 𝐹(𝑥) = (2.15) 𝜋2 1 2 −𝑥 e−2𝑥 e−3𝑥 + 𝑥 − [e − 2 + 2 − ⋯ ] , 𝑥 > 0. {6 2 2 3 2.4.3 Comparison Results Figures 2.9 (a) and 2.9(b) show QE as a function of laser wavelength calculated by the quantum model (Eq. 2.10), TSM (Eq. 2.12), FD model (Eq. 2.14), and Monte Carlo (MC) simulation based on TSM, for both copper and gold. In the calculation, the photon energy of the laser is chosen to be larger than the work function of the metal, such that the dominant electron emission mechanism is single-photon absorption. The laser field 𝐹1 in the quantum model is assumed to be 0.01V/nm. Though the absolute values for the QE from the quantum model are about one order smaller than those of the other models, their scalings are in remarkable agreement. In fact, by multiplying the QE from the quantum model with a constant 𝐶 (= 13.963 and 19.142 for copper and gold respectively), it matches the TSM and FD model results very well, as shown in Figs. 2.9a and 2.9b. 25 Figure 2.9 (a), (b) Quantum efficiency (QE) from quantum model (QM) (Eq. 2.10), Three-step model (TSM) (Eq. 2.12), Monte Carlo (MC) simulation based on TSM, and Fowler-DuBridge model (FD model) (Eq. 2.14), and the fitted QM × 𝐶 (= 13.963 and 19.142 for copper and gold respectively); (c), (d) QE from QM (Eq. 2.10), and TSM (Eq. 2.16) and FD model (Eq. 2.17) for parallel incidence. In all the calculations, we assume the dc field 𝐹0 = 0. For copper, 𝐸𝐹 = 7 eV and 𝑊0 = 4.31 eV [43], [44]; and for gold, 𝐸𝐹 = 5.53 eV and 𝑊0 = 5.1 eV [38], [91], [92]. In the QM, the laser field 𝐹1 = 0.01 V/nm. In both MC simulation and TSM, 𝐸𝑚 = 8.6 eV and the corresponding 𝜆𝑚 = 2.2 nm are used, for both copper [116], and gold [117]. In FD model, 𝑎1 = 5 × 10−18 m2/A [127] for copper, and 𝑎1 = 7 × 10−18 m2/A for gold (by fitting to TSM). The temperature is assumed at 𝑇 = 300 K. Difference between the quantum model and TSM as well as the FD model, however, is expected, due to the very different settings of the models. In the quantum model, the laser field is assumed to be perpendicular to the metal surface, whereas in the other models, the laser incidence (i.e., Poynting vector) is perpendicular to the metal surface. The laser power absorption in the latter case is expected to be significantly higher [128], [129], and therefore can induce more photoelectron emission. The case of laser field perpendicular to the metal surface may be realized when a linearly 26 polarized laser, of transverse electromagnetic (TEM) mode, propagates along the metal surface (i.e., parallel incidence). In this case, the laser power absorption coefficient is found to be 𝑃𝑎𝑏𝑠𝑜𝑟𝑝𝑡𝑖𝑜𝑛 /𝑃𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 = 𝜋𝛿𝑠 /𝜆 [128]–[130], where λ is the laser wavelength in vacuum, and 𝛿𝑠 = √2/𝜎𝜔𝜇0 is the skin depth, with σ being the conductivity of the metal, and 𝜇0 the vacuum permeability. By replacing (1 − 𝑅) with 𝜋𝛿𝑠 /𝜆 in Eqs. (2.12) and (14), the TSM and FD model for parallel incidence become, respectively, 2 𝜋𝛿𝑠 𝐸𝐹 + ℏ𝜔 𝐸𝐹 + 𝑊 𝑄𝐸 = ( ) 𝐹𝑒−𝑒 (1 − √ ) , TSM for parallel incidence, (2.16) 𝜆 2ℏ𝜔 𝐸𝐹 + ℏ𝜔 𝜋𝛿𝑠 ℏ𝜔 − 𝑊 𝑄𝐸 = 𝑎1 ( ) 𝐴𝑇 2 𝐹 ( ), FD model for parallel incidence, (2.17) 𝜆 𝑘𝐵 𝑇 where the resulted QE from both models is very close to that of the quantum model, Eq. (2.10), as shown in Figs. 2.9c and 2.9d. The difference between the quantum model and the other models for parallel incidence could be partially explained by the assumption in the quantum model that the laser field is terminated abruptly on the metal surface, where the finite penetration of the laser field and the electron-electron scattering effect inside the metal are ignored. These volume effects become important when the laser photon energy is larger than the work function of the metal [131]. It is important to note that the QE calculation from the quantum model in Fig. 2.9 is only for a relatively small laser field (0.01 V/nm). Larger laser field increases QE in the longer laser wavelength range, as already displayed in Fig. 2.8, which cannot be predicted from either TSM or the FD model. 27 Figure 2.10 Quantum efficiency (QE) from QM (Eq. 2.10) and TSM for parallel incidence (Eq. 2.16) under different dc fields 𝐹0 . The other parameters are the same as in Fig. 2.9. Figure 2.10 shows the comparison of QE from our quantum model and from the TSM for parallel incidence, for both copper and gold under small dc fields (i.e., 𝐹0 = 0.05 V/nm and 0.1 V/nm, respectively). The scaling of the results is in good agreement. Similar to Figs. 2.6 and 2.8, QE increases as dc field increases, especially in the longer laser wavelength, due to the combined effects of potential barrier lowering and narrowing by the dc bias. Both effects are captured in the quantum model, but only the barrier lowering effect (or Schottky effect) is included in the TSM model, leading to a higher QE from the quantum model. 2.5 Concluding Remarks In this chapter, we present a quantum model for photoemission from biased metal surfaces under the excitation of a perpendicular laser field, by solving the time-dependent Schrödinger equation exactly. The electron energy states distribution inside the metal are assumed to follow the Fermi-Dirac distribution. When the laser photon energy is larger than the work function of the metal, the electron emission process is dominated by single photon absorption. Electrons are predominantly emitted from the initial energy levels in the range within one photon energy below the Fermi level. When the laser field is small, QE is independent of the laser field strength. However, when the laser field increases ( ≥ 0.1 V/nm), QE increases with laser field strength in the longer laser wavelength range, which is due to the increased contributions from multiphoton absorption processes. This laser field- (or intensity-) dependent QE are not predicted by the pre-existing 28 photoemission models, such as the three-step model or Fowler-DuBridge model. It is also found that applying a dc field can increase photoelectron emission, especially in the longer laser wavelength range, which agrees with previous studies [38], [43], [44], [91], [92]. A comparison of QE by the quantum model and the existing classical models, i.e., the three- step model, the Fowler-DuBridge model and the Monte Carlo simulation, shows the scaling of the QE by those models agrees well, despite the very different settings and assumptions used in the quantum model and the classical models. The QE from the quantum model can be fitted to that of the three-step model by a constant of proportionality, which counts mainly for the different laser (or photon) absorption coefficient under different configurations. By using the laser power absorption coefficient for parallel incidence on the cathode surface, the QE from the classical models and the quantum model are in good agreement. We would like to emphasize again that the calculation presented in this chapter is only for a limited range of laser wavelengths when the photon energy is close to the work function of the metal, so that single photon absorption dominates the electron emission process. The work may be extended to the multiphoton emission and optical field emission regimes [38], [49], [132] in a much wider range of laser wavelengths. In these regimes, our model will be compared with the more general formalism of TSM [126] to examine the increasingly important effects of multiple scattering, transverse energies, and multiphoton absorption processes. For the single photon dominated electron emission studied here, it is found that higher temperature increases the emission current only in the longer wavelength ranges under steady state. It is important to note that, however, laser heating of electron gas and the cathodes is a time-dependent process. Also, high laser intensities (or laser fields) are often realized using femtosecond laser pulses without damaging the metal sample [132]–[134]. Due to the large difference between heat capacity of electrons and phonons, as well as picosecond-scaled electron-phonon scattering, there is a thermal nonequilibrium between electrons and the lattice [58], [135], [136], where the temperature of the electrons can reach up to thousands of kelvin [136], [137]. It also takes as long as ~1 ps for the electrons to reach a thermal equilibrium by electron-electron scattering, depending on the incident laser intensities [60]. Therefore, the electron system consists of both thermal electrons and non-thermal electrons [60], where the Fermi-Dirac electron distribution may not be valid. This in turn is expected to change the step- like characteristics in electron distribution (e.g., Fig. 2.5) by having the photo-excitation occur 29 from a significantly heated electron gas [136]. Consistent calculations of the time-evolved temperature of the electrons and lattice, electron distribution, and electron emission will be the subject of future study. 30 CHAPTER 3 EFFECTS OF LASER WAVELENGTH AND LASER HEATING ON PHOTOEMISSION FROM BIASED METAL SURFACES This chapter is based on the published journal paper “Quantum efficiency of photoemission from biased metal surfaces with laser wavelengths from UV to NIR,” J. Appl. Phys., vol. 130, no. 6, p. 064902, Aug. 2021, doi: 10.1063/5.0059497, by Yang Zhou and Peng Zhang. 3.1 Introduction Photoemission processes have been studied extensively over a wide range of laser wavelengths, both experimentally and theoretically. Single-photon photoemission process and the nonlinearity in strong laser intensity range due to the laser heating effect have been observed in the ultraviolet (UV) wavelength range for various metallic cathodes [41]–[44], [138]. Laser of wavelength around 800 nm has been widely used, and multiple photoemission mechanisms, including multiphoton absorption, above-threshold photoemission, photo-assisted field emission, and optical field tunneling [33], [38], [45], [91], [120], [139], [140] have been reported. With laser wavelengths from 1 μm to 1.5 μm, Park et al. [48] observed the narrowing of the emission cone angle of the fastest electrons when laser intensity increases, which is ascribed to field- induced steering of sub-cycle electrons. Few-cycle midinfrared (up to 8 μm) laser pulses are applied to single plasmonic nanotips, and it is found that electrons can escape the local field within a fraction of an optical half-cycle [49]. Single-cycle terahertz pulses have also been demonstrated to have their capacity to control nanotip photoemission electron dynamics [50]. Classical models, such as, the three-step model and the Fowler-DuBridge model [43], [44], [110]–[112], [114]–[117], mainly focus on single-photon (or photon energy ℏ𝜔 > work function 𝑊 , mostly in UV) photoemission. Fowler-DuBridge model has also been extended to include multiphoton emission and photo-thermionic emission processes [113], [141]–[144], however, the multiphoton emission coefficients typically require empirical fitting, and the model’s validity is questionable when applying to strong-field regimes [104]. Yalunin et al. [119] theoretically treat photoemission from metal surfaces, by perturbation theory, the Floquet method, and the Crank-Nicolson numerical approach. Despite extensive studies on photoemission by lasers of a wide wavelength range, there is still lack of systematic analysis on the effects of laser wavelength on photoemission and the corresponding quantum efficiency. In this chapter, we study the photoemission current and quantum efficiency over a wide range of laser wavelengths from 200 nm to 1200 nm, using a recent quantum model based on the 31 exact solution of time-dependent Schrödinger equation [125], which has been presented in chapter 2. We examine the laser heating effects on photoemission by using the two-temperature model (TTM) [58], [145], in which electrons and the lattice are treated in separate thermal equilibrium and are characterized by their own temperatures. TTM is found to be suitable for laser pulses on a time scale of a few hundreds of femtoseconds to tens of picosecond [41], [42], [146]–[149]. Our results show that the laser heating induced electron redistribution can enhance photoemission quantum efficiency, especially at laser wavelengths where the ratio of work function of cathode to photon energy is close to an integer. 3.2 Effects of Laser Wavelength on Photoemission Figure 3.1 shows the time-averaged transmission probability 𝑤𝑛 (𝜀 = 𝐸𝐹 ) through the nth channel (or n-photon process), under various combinations of dc electric field 𝐹0 , laser electric field 𝐹1 , and laser wavelength 𝜆 (= 2𝜋𝑐/𝜔). The metal is assumed to be gold, with Fermi energy of 𝐸𝐹 = 5.53 eV and work function of 𝑊0 = 5.1 eV. Unless specified otherwise, these are the default cathode properties in this chapter. The laser wavelengths are 200, 400, 800, and 1200 nm, corresponding to the photon energy of 6.20, 3.10, 1.55, and 1.03 eV, respectively. The laser fields 𝐹1 for lines in red, green, blue, and purple are 0.1, 1, 3, and 6 V/nm, corresponding to the local laser intensity of 1.33 × 109 , 1.33 × 1011 , 1.20 × 1012 , and 4.79 × 1012 W/cm2, respectively. The electron initial longitudinal energy is assumed to be at the Fermi level, i.e., 𝜀 = 𝐸𝐹 . It is obvious that the electron transmission probability 𝑤(𝐸𝐹 ) increases when the laser field increases. For 𝐹1 = 6 V/nm, there is more contribution from the large values of nth channel to the total transmission probability, especially for the cases of 𝜆 = 800 and 1200 nm. When 𝐹0 = 0, the dominant emission process for 𝜆 = 200, 400, 800, and 1200 nm are one-, two-, four-, and five-photon absorption processes, respectively. This is consistent with the integer value of 〈𝑊/ℏ𝜔 + 1〉, with 〈 〉 denoting the integer part of the expression. It is interesting to observe that the dominant channel shifts from 𝑛 = 5 to 𝑛 = 6 for 𝜆 = 1200 nm, when 𝐹1 increases to 3 and 6 V/nm (Fig. 3.1(d)), due to the channel closing effect [150], where the to-be-liberated electrons have to overcome not only the potential barrier but also the ponderomotive energy 𝑈𝑝 = 𝑒 2 𝐹12 /4𝑚𝜔2 in the laser field [33] such that the electron drift kinetic energy 𝐸𝑛 = 𝜀 + 𝑛ℏ𝜔 − 𝐸𝐹 − 𝑊 − 𝑈𝑝 is larger than zero. A small dc field 𝐹0 = 0.1 V/nm has little effect on the transmission probability for 𝜆 = 200 and 400 nm (Figs. 3.1(e) and 3.1(f)). However, it shifts the 32 dominant channel from 𝑛 = 4 to 𝑛 = 3 for 𝜆 = 800 nm when 𝐹1 = 0.1, 1, and 3 V/nm (Fig. 3.1(g)), and also shifts the dominant channel from 𝑛 = 6 to 𝑛 = 5 for 𝜆 = 1200 nm when 𝐹1 = 3 and 6 V/nm (Fig. 3.1(h)). This is due to the lowering of the surface potential barrier by the dc electric field, and therefore lowering the number of photons required to overcome the barrier. When 𝐹0 = 1 V/nm, more nth channels of lower order are opened up, and the dominant channel is further shifted to smaller value of nth channels, especially for 𝜆 = 400, 800, and 1200 nm (Figs. 3.1(j) – 3.1(l)). When 𝐹0 = 5 V/nm, the dominant emission process is direct field tunneling, i.e., through 𝑛 = 0, regardless of the laser wavelength. Figure 3.1 Electron transmission probability from 𝜀 = 𝐸𝐹 through nth channel, under various combinations of dc electric field 𝐹0 , laser field 𝐹1 and laser wavelength 𝜆. (a) – (d) 𝐹0 = 0 V/nm; (e) – (h) 𝐹0 = 0.1 V/nm; (i) – (l) 𝐹0 = 1 V/nm; and (m) – (p) 𝐹0 = 5 V/nm. In each column, (a), (e), (i), (m) 𝜆 = 200 nm; (b), (f), (j), (n) 𝜆 = 400 nm; (c), (g), (k), (o) 𝜆 = 800 nm; and (d), (h), (l), (p) 𝜆 = 1200 nm. 𝐹1 corresponding to lines in red, green, blue, and purple, are 0.1, 1, 3, and 6 V/nm, respectively. The metal is gold, with 𝐸𝐹 = 5.53 𝑒𝑉 and 𝑊0 = 5.1 𝑒𝑉. 33 Figure 3.2 Electron transmission probability 𝐷(𝜀) as a function of electron initial energy 𝜀 and laser wavelength 𝜆, for various combinations of laser field 𝐹1 and dc electric field 𝐹0 . (a) – (d) 𝐹0 = 0 V/nm; (e) – (h) 𝐹0 = 1 V/nm; (i) – (l) 𝐹0 = 5 V/nm. In each column, (a), (e), (i) 𝐹1 = 0.1 V/nm; (b), (f), (j) 𝐹1 = 1 V/nm; (c), (g), (k) 𝐹1 = 3 V/nm; (d), (h), (l) 𝐹1 = 6 V/nm. The electron transmission probability 𝐷(𝜀) as a function of electron longitudinal energy 𝜀 and laser wavelength 𝜆, under various combinations of dc electric field 𝐹0 and laser field 𝐹1 , is shown in Fig. 3.2. When 𝐹0 = 0 , the three-dimensional (3D) surface plot shows stair-like behavior as a function of 𝜀 for a given 𝜆 (Figs. 3.2(a)–3.2(d)). Each stair corresponds to n-photon absorption process, with n increasing as 𝜀 decreases, for a given 𝜆. As 𝐹1 increases, the electron transmission probability increases and more “stairs” clearly appear at the bottom right region, i.e., the “smaller initial energy”-“longer laser wavelength” region. When 𝐹0 = 1 V/nm, the electron transmission probability is increased due to the lowering of the surface potential barrier. The step edge between “stairs” is not as steep as that with zero dc field but becomes gradual (Figs. 3.2(e)‒3.2(h)). Dc electric field has a greater enhancement on the emission induced by a longer wavelength laser [125]. This can be explained by the change of the dominant emission process, which shifts to smaller n-photon absorption process due to the lowering of the surface potential barrier by the applied dc field. From Figs. 3.1(a)–3.1(d), it can be observed that the (n+1)-photon absorption process has a probability orders of magnitude lower than the n-photon absorption 34 process, for a given 𝜆. When 𝐹0 = 5 V/nm, the 3D surface plot of 𝐷(𝜀) becomes smooth (Figs. 3.2(i)-3.2(l)). At each electron initial longitudinal energy 𝜀 , 𝐷(𝜀) is almost constant for all different laser wavelengths, which implies that the dominant emission process is due to the dc field tunneling, consistent with Figs. 3.1(m)-3.1(p). In Figs. 3.2(i)–3.2(l), as the laser field 𝐹1 increases from 0.1 V/nm to 6 V/nm, 𝐷(𝜀) increases and the stair shape of the surface in the shorter wavelength range becomes more obvious. The shape of the 3D surfaces is determined by the relative strength of laser and dc electric fields. Figure 3.3 Electron transmission probability 𝐷(𝜀 = 𝐸𝐹 ) with the initial energy of 𝐸𝐹 , as a function of laser wavelength 𝜆, for various combinations of dc electric field 𝐹0 and laser field 𝐹1 . (a) 𝐹0 = 0 ; (b) 𝐹0 = 0.1 V/nm; (c) 𝐹0 = 1 V/nm; (d) 𝐹0 = 5 V/nm. The laser fields 𝐹1 corresponding to solid lines in red, green, blue, purple, and the dashed line in cyan, are 0.1, 1, 3, 6, and 0 V/nm, respectively. Figure 3.3 shows the electron transmission probability 𝐷(𝜀 = 𝐸𝐹 ) as a function of laser wavelength under various combinations of dc and laser electric fields, which is the projection of 𝐷(𝜀) in the 𝐷 − 𝜆 plane with 𝜀 = 𝐸𝐹 in Fig. 3.2. When 𝐹0 = 0, the curves display distinct stair shape, though each step is not flat, especially for 𝜆 > 486 nm, i.e., 𝑛 = 𝑊0 /ℏ𝜔 > 2. For small 𝐹1 , 𝐷(𝐸𝐹 ) has its maximum value at the step point in each step, corresponding to an integer value of the ratio of 〈𝑊0 /ℏ𝜔 + 1〉. As the laser field 𝐹1 increases, 𝐷(𝐸𝐹 ) is enhanced greatly, especially for 𝜆 > 486 nm. This is because 𝐷(𝜀 = 𝐸𝐹 ) ∝ 𝐹12𝑛 in the n-photon photoemission 35 regime. Meanwhile, the step points shift to smaller laser wavelength when 𝐹1 gets larger, which is indicated by the dashed arrow line in Fig. 3.3(a). This shift is due to the channel closing effect, which is more pronounced for longer laser wavelengths. When there is a dc electric field but small (𝐹0 = 0.1 V/nm), the “stair” becomes smoother. The step points shift to larger wavelength (i.e., smaller photon energy) due to the lowering of the surface potential barrier by the applied dc electric field. As 𝐹0 increases to 1 V/nm, the “stairs” still exist but the number of the “stairs” decreases and the magnitude difference between them gets smaller. When 𝐹1 = 0.1 V/nm, there are two distinct “stairs” observed. Most of the emission is due to photon-assisted tunneling (cf. Figs. 3.1(k) and 3.1(l)) for relatively long laser wavelength. When 𝐹0 = 5 V/nm, the electron transmission probability becomes almost independent of laser wavelength for 𝐹1 = 0.1 V/nm. The dashed curve is for dc field emission without laser electric field with 𝐹0 = 5 V/nm, which overlaps with the curve for 𝐹1 = 0.1 V/nm. For 𝐹1 ≥ 1 V/nm, the dominant emission process is still direct tunneling, as shown in Figs. 3.1(m) – 3.1(p). The laser electric field 𝐹1 can modulate the emission process through photo-assisted field emission and above-threshold photoemission. It can be observed that, with a large dc field, 𝐷(𝐸𝐹 ) is larger in the relatively longer laser wavelength range, as shown in Fig. 3.3(d). The emission current density 𝐽, calculated from Eq. (2.9), as a function of laser wavelength 𝜆, is presented in Fig. 3.4. It should be pointed out that the laser heating effect is not considered here, with the temperature 𝑇 in Eq. (2.9) set to 300 K. Figure 3.4 shares similar trends as in Fig. 3.3, since the majority of emitted electrons originate from the vicinity of the Fermi level [33], [125]. However, the curves for 𝐽 vs 𝜆 are smoother than those for 𝐷(𝜀 = 𝐸𝐹 ) vs 𝜆, due to the combined emitted electrons with different initial longitudinal energies. Including a dc field will make the step edge smoother and the step point shift towards longer laser wavelength. When 𝐹0 = 5 V/nm, the electron emission is increased by 1 ~ 21 orders of magnitude compared with the case of no applied dc electric field, for different combinations of laser wavelength 𝜆 and laser field 𝐹1 . The dominant emission process of field emission (𝑛 = 0) for 𝐹0 = 5 V/nm makes the emission current insensitive to the laser wavelength. 36 Figure 3.4 Electron emission current density, calculated from Eq. (2.9) as a function of laser wavelength 𝜆, for various combinations of dc electric field 𝐹0 and laser field 𝐹1 . The temperature 𝑇 is set to be constant at 300 K, without considering laser heating. (a) 𝐹0 = 0; (b) 𝐹0 = 0.1 V/nm; (c) 𝐹0 = 1 V/nm; (d) 𝐹0 = 5 V/nm. The laser fields 𝐹1 corresponding to solid lines in red, green, blue, purple, and the dashed line in cyan, are 0.1, 1, 3, 6, and 0 V/nm, respectively. Figure 3.5 shows quantum efficiency (QE) as a function of laser wavelength 𝜆 under various laser field strengths 𝐹1 and dc field strengths 𝐹0 , which is calculated from Eq. (2.10). It is obvious that the curves for 𝑄𝐸 vs 𝜆 shares similar trends as those for 𝐽 vs 𝜆 in Fig. 3.4. For 𝐹0 ≤ 1 V/nm, QE is almost independent of 𝐹1 for different laser fields 0.1 V/nm ≤ 𝐹1 ≤ 6 V/nm in the smaller laser wavelength regime (or 𝑊/ℏ𝜔 < 1), which is indicated by the yellow shaded area in Figs. 3.5(a)-(c). This is because the dominant single-photon photoemission in this regime follows 𝐽 ∝ 𝐼. Therefore, the quantum efficiency is independent of the laser field as 𝑄𝐸 ∝ 𝐽/𝐼. Note the majority of existing studies on QE focus only on this single-photon regime [43], [44], [125]. For 𝑊0 /ℏ𝜔 > 1, the photoemission is due to n-photon process, which scales as 𝐽 ∝ 𝐼 𝑛 (𝑛 ≥ 2), with 𝑛 determined by rounding 𝑊/ℏ𝜔 up to the nearest integer number. As a result, 𝑄𝐸 ∝ 𝐼 𝑛−1 , and the quantum efficiency increases greatly as 𝐹1 increases. As dc field increases, QE is enhanced and the step points are shifted towards longer laser wavelengths (i.e., smaller photon energy), due to the lowering of the surface potential barrier by the dc field. As a result, increasing 𝐹0 from 0 to 1 V/nm, also extends the single-photon absorption regime from ~250 nm 37 to ~320 nm, where QE is independent of laser field 𝐹1 , as indicated by the yellow shaded area in Figs. 3.5(a)-(c). When 𝐹0 = 5 V/nm (Fig. 3.5(d)), QE can be larger than 1, due to the dominant contribution of dc field emission. Figure 3.5 Quantum efficiency (QE), calculated from Eq. (2.10) as a function of laser wavelength 𝜆, for various combinations of dc electric field 𝐹0 and laser field 𝐹1 . The temperature is set to be constant at 300 K, without considering laser heating. (a) 𝐹0 = 0; (b) 𝐹0 = 0.1 V/nm; (c) 𝐹0 = 1 V/nm; (d) 𝐹0 = 5 V/nm. The laser fields 𝐹1 corresponding to solid lines in red, green, blue, and purple are 0.1, 1, 3, and 6 V/nm, respectively. 3.3 Laser Heating Effects 3.3.1 Two-Temperature Model It is known that electron energy distribution function (EEDF) and its dynamic due to laser heating have strong influence on photoelectron emission current and emission electron energy spectrum [41], [42], [63], [125], [145]. Electrons can be excited to higher energy levels by absorbing the laser energy. The excited electrons can come into thermal equilibrium with other electrons by electron-electron scatterings, and can transfer energy to the lattice by electron- phonon scatterings. The microscopic kinetic approach, such as Boltzmann’s equation, can provide an accurate estimation of internal thermalization process and electron and phonon energy distribution, especially for femtosecond laser pulses [62], [120], [151]. However, the classical two-temperature model (TTM) still works well for a laser pulse of ~500 fs duration [41], [42], 38 [146]–[149], but with much lower computational complexity. We use TTM to estimate the laser heating effects in photoemission. In TTM [58], [135], electrons and lattice are considered as two separate equilibrium subsystems, characterized by their own temperatures 𝑇𝑒 and 𝑇𝑙 , 𝜕𝑇𝑒 (𝑥, 𝑡) 𝜕 𝜕𝑇𝑒 (𝑥, 𝑡) 𝐶𝑒 (𝑇𝑒 ) = 𝜅 − 𝑔(𝑇𝑒 − 𝑇𝑙 ) + 𝐺(𝑥, 𝑡), (3.1𝑎) 𝜕𝑡 𝜕𝑥 𝜕𝑥 𝜕𝑇𝑙 (𝑥, 𝑡) 𝐶𝑙 (𝑇𝑙 ) = 𝑔(𝑇𝑒 − 𝑇𝑙 ). (3.1𝑏) 𝜕𝑡 7 𝜋𝑘𝐵 𝑇𝑒 2 In Eq. (3.1), the electron heat capacity 𝐶𝑒 (𝑇𝑒 ) = 𝛾𝑇𝑒 / [1 + 40 ( ) ] with 𝑘𝐵 being the 𝐸𝐹 1 𝑚𝜂 Boltzmann’s constant, 𝛾 = 3 𝜋 2 𝑘𝐵2 𝐵(𝜀 = 𝐸𝐹 ) with 𝐵(𝜀) = 8𝜋 (2𝜋ℏ)3 √2𝑚𝜀 and 𝜂 an effective thermal mass term. When 𝑘𝐵 𝑇𝑒 ≪ 𝐸𝐹 , 𝐶𝑒 (𝑇𝑒 ) ≈ 𝛾𝑇𝑒 . The lattice heat capacity 𝐶𝑙 (𝑇𝑙 ) = 1 𝑇 2 3𝑁𝑘𝐵 / [1 + 20 ( 𝑇𝐷) ] , where 𝑁 is the number density of the atoms in metal, 𝑇𝐷 = 𝑙 ℏ𝑣𝑠 (6𝜋 2 𝑁𝑟)1/3 is the Debye temperature, with 𝑣𝑠 the speed of sound inside the metal and 𝑟 the 𝑘𝐵 𝜋𝜆0 number of atoms per unit cell. When 𝑇𝐷 ≪ 𝑇𝑙 , 𝐶𝑙 is a constant. 𝑔 = 𝑘𝐵 𝑘𝐹3 𝑚𝑣𝑠2 is the electron- 9ℏ lattice coupling coefficient, where 𝑘𝐹 is determined by the Fermi momentum ℏ𝑘𝐹 = √2𝑚𝐸𝐹 , 𝜆0 is a dimensionless electron-phonon coupling constant, which is characteristic of the metal and on the order of 0.1 to 1. 𝑔(𝑇𝑒 − 𝑇𝑙 ) gives the amount of energy per unit volume per unit time transferred between electron and lattice systems. 𝐺(𝑥, 𝑡) = 𝐼(𝑡)𝑃𝑎𝑏𝑠 𝛼 exp(−𝛼𝑥) is the energy absorbed by the metal, where 𝐼(𝑡) is the laser intensity temporal profile, 𝛼 = 4𝜋𝑘/𝜆 is the absorption coefficient with 𝑘 the extinction coefficient, 𝑃𝑎𝑏𝑠 = 𝜋𝛿𝑠 /𝜆 is the laser power absorption coefficient for the parallel-to-surface incident laser [128]–[130], with 𝜆 the laser wavelength in the vacuum and 𝛿𝑠 = √2/𝜎𝜔𝜇0 the skin depth, 𝜎 the conductivity of the metal, and 𝜇0 the vacuum permeability, or 𝑃𝑎𝑏𝑠 = 1 − 𝑅 for the incident laser tilted to the metal surface, 2𝐸𝐹 with 𝑅 the reflectivity of the laser at the metal surface. 𝜅 = 𝐶𝑒 (𝑇𝑒 )𝜏 is the thermal 3𝑚 conductivity, with 𝜏 the electron scattering time and 𝑚 the electron rest mass. According to 1 1 1 ℏ𝐸𝐹 1 ℏ 1 Matthiessen’s rule, =𝜏 +𝜏 , with 𝜏𝑒−𝑒 = 𝐴 2 2 and 𝜏𝑒−𝑝ℎ = 2𝜋𝑘 the electron- 𝜏 𝑒−𝑒 𝑒−𝑝ℎ 0 𝑘𝐵 𝑇𝑒 𝐵 𝜆0 𝑇𝑖 electron and electron-phonon scattering times respectively, and 𝐴0 a dimensionless, material- specific quantity. 39 3.3.2 Electron Temperature 𝑻𝒆 The time-dependent evolution of the electron temperature in response to an incident laser pulse with full width at half maximum (FWHM) 𝜏𝐹𝑊𝐻𝑀 = 450 fs under various laser wavelengths and field strengths is shown in Figs. 3.6(a) and 3.6(b). In Fig. 3.6(a), the laser field strength is kept constant as 𝐹1 = 1 V/nm for laser wavelengths ranging from 200 nm to 1200 nm. The maximum of the electron temperature lags behind that of the incident laser intensity (dotted curve), which is centered at 𝑡 = 0. It is interesting to find that as the laser wavelength decreases, the time delay between the maximum of the temperature and that of the laser intensity increases, and the temperature gets larger, which is due to larger laser energy absorption 𝐺~𝑘/𝜆3/2 inside the metal for parallel incidence (extinction coefficient 𝑘 vs 𝜆 shown in Fig. 3.6(c)). For a fixed laser wavelength (𝜆 = 800 nm), the electron temperature increases with the laser field strengths, as shown in Fig. 3.6(b). In addition, the time delay between the maximum of the electron temperature and that of the laser intensity increases. The electron temperature at 𝑡 = 0, which peak corresponds to the peak of the laser intensity, and the peak electron temperature 𝑇𝑒 , as a function of laser wavelength are shown in Fig. 3.6(d) for 𝐹1 = 1 V/nm. The temperature decreases with the laser wavelength, which is due to the increased time delay and less absorption of the laser energy as laser wavelength increases. The obtained electron temperature 𝑇𝑒 is then used in Eq. (2.9) by setting 𝑇 = 𝑇𝑒 to calculate the photoemission current density. 40 Figure 3.6 Electron temperature varies with incident laser wavelength 𝜆 and laser field strength 𝐹1 calculated from Eq. (3.1). (a) Electron temperature temporal profile for 𝜆 from 200 nm to 1200 nm with peak laser field 𝐹1 = 1 V/nm and the corresponding peak laser intensity of 1.33 × 1011 W/cm2. The dotted curve is the laser intensity temporal profile I(t). (b) Electron temperature temporal profile for peak laser field 𝐹1 from 0.2 V/nm to 1 V/nm with 𝜆 = 800 nm. (c) Optical constant extinction coefficient k as a function of 𝜆 [152]. (d) Electron temperature at 𝑡 = 0 (black curve), which corresponds to the time instant of peak laser intensity, and peak peak electron temperature 𝑇𝑒 (red curve), as a function of 𝜆. Parameters in Eq. (3.1) for gold are 𝜂 = 1.08, 𝐶𝑙 = 2.35 × 106 J/m3 /K (assumed constant), 𝑣𝑠 = 3240 m/s, 𝜎 = 4.11 × 107 S/m, 𝐴0 = 17, 𝜆0 = 0.1548 [41], [145]. Erratum: in ref. [153], x-axis label should be t (fs) instead of t (ps). 3.3.3 Current Density and Quantum Efficiency with Laser Heating A comparison of photoemission with laser heating effects (red curves) and without laser heating effects (blue curves, with 𝑇 = 𝑇𝑒 ≡ 300K) is shown in Fig. 3.7. The results are taken at t = 0, which corresponds to the laser intensity peak (cf. the dotted curve in Fig. 3.6(a)). The peak laser field strength is taken to be 1 V/nm. The electron emission current density per electron initial energy, calculated as 𝐽(𝜀) = 𝑒𝐷(𝜀)𝑁(𝜀) from Eq. (2.10), extends to energy levels above the Fermi level as the laser heating effect is considered, as shown in Fig. 3.7(a) for 𝜆 = 200 nm, 41 600 nm, and 1000 nm. This is because more electrons are thermalized to energy levels above the Fermi level by absorbing the laser energy. The electron emission from initial energy levels above the Fermi energy accounts for 7.84%, 6.85%, and 91.1% of the total emission for 𝜆 = 200 nm, 600 nm, and 1000 nm, respectively, with laser heating effects, compared to 0.13%, 0.015%, and 0.73% without laser heating. Figure 3.7(b) shows the emission current density as a function of the laser wavelength. The steep step point, which is the transition point between different 𝑛- photon processes, becomes smooth. This is because more electrons above the Fermi level can be emitted through smaller 𝑛 -photon process, which has an emission probability orders of magnitude higher than larger 𝑛-photon process (see Figs. 3.1(a)-(d) and its description in section 3.2). The difference between the red and blues curves shows that laser heating effect has a greater impact on longer wavelength laser-induced photoemission. The quantum efficiency as a function of laser wavelength is shown in Fig. 3.7(c), showing the same trend as 𝐽 vs 𝜆 in Fig. 3.7(b). In summary, the increase of QE due to laser heating is the strongest near the step points (i.e., 𝑊0 /ℏ𝜔 = integer) and is more profound for longer laser wavelengths. Figure 3.7 Laser heating effects on photoemission. (a) Electron emission current density per electron initial energy at 𝑡 = 0 for 𝜆 = 200 nm, 600 nm, and 1000 nm, with 𝐹1 = 1 V/nm and 𝐹0 = 0; (b) Electron emission current density and (c) QE at 𝑡 = 0 as a function of laser wavelength for 𝐹1 = 1 V/nm and zero dc field 𝐹0 = 0. 42 3.3.4 Comparison with Experimental Results Figure 3.8 Comparison with experimental results. (a) Calculated emission current density temporal profile for various laser intensities used in the experiment [41]. (b) Emission current density as a function of the peak laser intensity. Scatters are experimental data extracted from [41]. The red curve is calculated from Eq. (3.2) with 𝐽(𝑡) calculated by our quantum model. (c) Quantum efficiency as a function of the peak laser intensity. Parameters in Eq. (3.1) for copper are 𝛾 = 96.6 J/K 2 /m3 , 𝐶𝑙 = 3.5 × 106 J/m3 /𝐾 (assumed constant), 𝑣𝑠 = 5010 m/s , 𝜎 = 5.96 × 107 S/m , 𝐴0 𝑘𝐵2 /ℏ𝐸𝐹 = 2.3 × 107 /(K 2 s) , 2𝜋𝑘𝐵 𝜆0 /ℏ = 1.1 × 1011 /(K 2 s) [41], [42], [145]. We demonstrate the validity of the above quantum model with laser heating, by comparing it with experimental results in Ref. [41]. In the experiment, a laser pulse of 450 fs duration at 248 nm is used. The metal is copper, with Fermi energy 𝐸𝐹 = 7 eV and work function 𝑊0 = 4.6 eV. The emission current density per pulse based on our quantum model [93], [125] is given as, ∞ 1 𝐽𝑝 = ∫ 𝐽(𝑡)𝑑𝑡 , (3.2) 𝜏𝐹𝑊𝐻𝑀 −∞ where 𝐽(𝑡) is the emission current density at 𝑡 calculated from Eqs. (2.8) and (2.9), and 𝜏𝐹𝑊𝐻𝑀 is the full-width-half-maximum of the intensity of the laser pulse. Note that though Eq. (2.8) is the time-averaged transmission probability for continuous wave laser excitation, it is found to be an excellent approximation for laser pulses of longer than 10 cycles [38]. Thus, it is expected to be applicable for laser pulses of 450 fs (~ 544 laser cycles) at 248 nm considered here. The temporal profile of the emission current density is shown in Fig. 3.8(a). As the laser intensity increases, the emission current density increases, and the current density peak lags behind the laser intensity peak. This is due to the delay between the temperature peak and the laser intensity peak, 43 as shown in Figs. 3.6(a) and 3.6(b), which has also been observed in [154]. The calculated current density by the quantum model from Eq. (3.2) is shown as red curve in Fig. 8(b), which is in good agreement with the experimental measured current density shown as blue scatter points in Fig. 3.8(b). The small difference can be ascribed to the different settings of the experiment and our quantum model. In the experiment, the laser field is incident onto the metal surface with an angle of 80° to the normal. However, in our model, the laser field is perpendicular to metal surface (i.e., parallel incidence). For n-photon absorption process, the emission current density 𝐽 ∝ (𝐹1 cos 𝜃)2𝑛 with 𝜃 being the angle between field polarization and the normal of the cathode surface [139]. Therefore, our model slightly overestimates the photoemission. The quantum efficiency is plotted in Fig. 3.8(c). To be consistent with the current density Eq. (3.2), the laser ∞ intensity I in Eq. (2.10) is calculated by 𝐼 = ∫−∞ 𝐼(𝑡) 𝑑𝑡/𝜏𝐹𝑊𝐻𝑀 in order to calculate QE. It is clear that QE increases with the laser intensity instead of being constant (yellow shaded area) as shown in Figs. 3.5(a)–3.5(c), which is ascribed to the laser heating induced electron redistribution. 3.4 Concluding Remarks In this chapter, we have analyzed photoemission from metal surfaces with the laser wavelength from 200 nm to 1200 nm (i.e., UV to NIR), based on an analytical quantum model by solving the time-dependent Schrödinger equation. The photoemission mechanisms vary from multiphoton absorption to dc or optical field emission, depending on the laser wavelength and intensity, and dc electric field. When 𝐹0 ≤ 0.1 V/nm, which is much smaller than the typical dc field used in static field emission, the emission current density and quantum efficiency (QE) are characterized by different n-photon absorption processes. Channel closing effects and more above-threshold photoemission (𝑛 > 𝑊/ℏ𝜔) are observed as the laser field increases, especially for longer wavelengths. It is found that QE in the short wavelength regime (or single-photon regime, n = 1) is insensitive to the laser field strength for 𝐹0 ≤ 1 V/nm. When 𝐹0 = 5 V/nm, the static field emission (n = 0) becomes dominant, regardless of the laser wavelength (200 nm ≤ 𝜆 ≤ 1200 nm) and the laser field strength (0.1 V/nm ≤ 𝐹1 ≤ 6V/nm). Laser heating induced electron dynamics is considered by using two-temperature model (TTM), which is applicable for sub-picosecond laser pulses. The electron temperature rise shows a strong dependence on the laser wavelength. It is found that QE increases nonlinearly with the laser intensity for sub-picosecond laser pulses. The increase is the strongest near wavelengths 44 where the work function of the metal is integer multiple of the corresponding photon energy. The quantum model with the laser heating included also reproduces previous experimental results, which further validates our quantum model and the importance of laser heating. 45 CHAPTER 4 FIELD EMISSION FROM METAL SURFACES WITH NANOSCALE DIELECTRIC COATINGS This chapter is based on the published journal paper “Theory of field emission from dielectric coated surfaces,” Phys. Rev. Research, vol. 2, no. 4, p. 043439, Dec. 2020, doi: 10.1103/PhysRevResearch.2.043439, by Yang Zhou and Peng Zhang. 4.1 Introduction Electron field emission [97], [106] attracts intensive attention in many applications, such as, flat panel display [155]–[157], electron microscopes [19], [158], [159], vacuum micro- electronics [160], [161] and emerging nano-devices [27], [28], [109], [162], [163], X-ray sources [164], and high power microwave sources and amplifiers [165]–[172], for its high brightness, low emittance and miniaturized device size [173], [174]. Common challenges of field emission include the operation requirement of high vacuum condition and current instabilities [175], [176]. To overcome these problems, ultra-thin coatings, such as graphene, graphene oxide and zinc oxide [70], [177]–[179], are fabricated onto the emitter to provide chemical and mechanical protection. Coated emitters are demonstrated to not only have longer current stability, but also smaller turn-on electric field (i.e., field at which the cathode starts appreciable electron emission) and enhanced emission current due to the lowering of the effective potential barrier [47], [70], [178]–[181]. In addition to the artificially fabricated coatings, native oxides or foreign adsorbates can be easily formed on the surface of the emitter at low vacuum condition [182]. The thin oxide film or the coated dielectric layer on the cathode surface forms a double-layer potential barrier, which strongly influences the field emission properties. The heterostructure in the emission barrier introduced by the coating also has its potential to change the electrons’ mean transverse energy behavior that affects beam quality, which makes it an active area for photoinjectors for future X-ray free electron lasers (XFELs) [69], [78]. A modified Fowler-Nordheim equation was constructed to account for the double-barrier field emission scenario [183], [184]. However, there is still lack of systematic analysis on the parametric scaling of field emission from coated surfaces and comprehensive understanding on the interplay of various parameters to optimize the design of coated field emitters. In this chapter, we develop a quantum analytical solution for field emission from the dielectric-coated cathode surface, by solving the one-dimensional (1D) time-independent Schrödinger equation subject to the double-barrier introduced by the coating layer. The solution 46 is applicable for arbitrary electric dc field, cathode properties (i.e., work function and Fermi level) and dielectric coating properties (i.e., dielectric constant, electron affinity and thickness). It includes not only field emission but also thermionic emission. The model predicts that for 1D flat surfaces, coatings of small dielectric constant and large electron affinity tend to enhance the field emission current, which provides insights for the design of a stable and efficient field emitter. 4.2 Analytical Model Figure 4.1 Field emission from a metal surface coated with a dielectric. The metal-dielectric interface is located at x = 0, and the coating’s thickness is d. The metal has Fermi level 𝐸𝐹 and work function 𝑊. The dielectric has electron affinity of 𝜒 and dielectric constant of 𝜀𝑑𝑖𝑒𝑙 . The electron initial longitudinal energy is 𝜀. The external dc field of 𝐹 (in the vacuum) is applied to the emitter surface. The field in the dielectric is 𝐹𝑑𝑖𝑒𝑙 = 𝐹/𝜀𝑑𝑖𝑒𝑙 . The energy diagram for electron emission from a 1D dielectric-coated metal surface is shown in Fig. 4.1. Electrons inside the metal would see a double-triangular potential barrier, 0, 𝑥 < 0, 𝑉(𝑥) = { 0𝑉 − 𝜒 − 𝑒𝐹 𝑑𝑖𝑒𝑙 𝑥, 0 ≤ 𝑥 < 𝑑, (4.1) 𝑉0 + 𝑒(𝐹 − 𝐹𝑑𝑖𝑒𝑙 )𝑑 − 𝑒𝐹𝑥, 𝑥 ≥ 𝑑, where 𝑉0 = 𝑊 + 𝐸𝐹 , with 𝑊 and 𝐸𝐹 being the work function and Fermi level of the metal, respectively; 𝜒 is the electron affinity of the dielectric; 𝑒 is the elementary charge (positive); 𝐹 is the applied dc electric field in the vacuum; 𝐹𝑑𝑖𝑒𝑙 = 𝐹/𝜀𝑑𝑖𝑒𝑙 is the dc electric field inside the dielectric with 𝜀𝑑𝑖𝑒𝑙 the dielectric constant of the coating layer; and 𝑑 is the thickness of the dielectric coating layer. To calculate the probability of electron tunneling through the barrier, we solve the 1D time- independent Schrödinger equation, 47 ℏ2 𝜕 2 − 𝜓(𝑥) + [𝑉(𝑥) − 𝜀]𝜓(𝑥) = 0, (4.2) 2𝑚 𝜕𝑥 2 where 𝜓(𝑥) is the complex electron wave function, ℏ is the reduced Planck’s constant, 𝑚 is the electron mass, 𝑉(𝑥) is the potential given in Eq. (4.1), and 𝜀 is the initial longitudinal energy of electrons incident on the metal surface. Here, for simplicity, the electron mass 𝑚 in all the three regions (i.e., metal, dielectric, and vacuum) is set equal to the electron rest mass. For 𝑥 < 0, the solution to Eq. (4.2) is 𝜓𝐼 (𝑥) = 𝑒 𝑖𝑘0 𝑥 + 𝑅1 𝑒 −𝑖𝑘0 𝑥 , (4.3) where 𝑘0 = √2𝑚𝜀/ℏ2 , 𝑅1 is the reflection coefficient at the metal-dielectric interface. Equation (4.3) represents the superposition of an incident wave and a reflected wave. For 0 ≤ 𝑥 < 𝑑, Eq. (4.2) can be solved by transforming it into the form of 𝑑 2 𝜓/𝑑𝜂12 + 𝜂1 𝜓 = 0 [97], [185]–[187], and the solution is expressed in terms of Airy functions as 𝜓𝐼𝐼 (𝑥) = 𝑎𝐴𝑖(−𝜂1 ) + 𝑏𝐵𝑖(−𝜂1 ), (4.4) 2𝑚𝑒𝐹𝑑𝑖𝑒𝑙 1/3 𝜀−𝑉1 where 𝜂1 = ( ) (𝑥 + 𝑒𝐹 ), with 𝑉1 = 𝑉0 − 𝜒, 𝐴𝑖 and 𝐵𝑖 are the Airy functions of the ℏ2 𝑑𝑖𝑒𝑙 first kind and second kind, respectively. 𝜓𝐼𝐼 represents the superposition of the transmitted wave from the metal-dielectric interface and the reflected wave from the dielectric-vacuum interface. For 𝑥 ≥ 𝑑, the Schrödinger equation, Eq. (4.2), is transformed into 𝑑 2 𝜓/𝑑𝜂22 + 𝜂2 𝜓 = 0, whose solution is 𝜓𝐼𝐼𝐼 (𝑥) = 𝑇3 [𝐴𝑖(−𝜂2 ) − 𝑖𝐵𝑖(−𝜂2 )], (4.5) 2𝑚𝑒𝐹 1/3 𝜀−𝑉2 where 𝜂2 = ( ) (𝑥 + ), with 𝑉2 = 𝑉0 + 𝑒(𝐹 − 𝐹𝑑𝑖𝑒𝑙 )𝑑. Equation (4.5) represents an ℏ2 𝑒𝐹 outgoing wave traveling into the vacuum. The imposition of the boundary conditions that 𝜓 and 𝑑𝜓/𝑑𝑥 are continuous at both the metal-dielectric interface 𝑥 = 0 and the dielectric-vacuum interface 𝑥 = 𝑑 gives 1 + 𝑅1 = 𝑎𝐴1 + 𝑏𝐵1 , (4.6𝑎) 1 − 𝑅1 = 𝜁(𝑎𝐴1′ + 𝑏𝐵1′ ), (4.6𝑏) 𝑎𝐴2 + 𝑏𝐵2 = 𝑇3 (𝐴3 − 𝑖𝐵3 ), (4.6𝑐) 𝑎𝐴′2 + 𝑏𝐵2′ = 𝜉𝑇3 (𝐴′3 − 𝑖𝐵3′ ), (4.6𝑑) where 𝐴1 = 𝐴𝑖(−𝜂1 (𝑥 = 0)) , 𝐵1 = 𝐵𝑖(−𝜂1 (𝑥 = 0)) , 𝐴1′ = 𝐴𝑖 ′ (−𝜂1 (𝑥 = 0)) , 𝐵1′ = 𝐵𝑖 ′ (−𝜂1 (𝑥 = 0)) , 𝐴2 = 𝐴𝑖(−𝜂1 (𝑥 = 𝑑)) , 𝐵2 = 𝐵𝑖(−𝜂1 (𝑥 = 𝑑)) , 𝐴′2 = 𝐴𝑖 ′ (−𝜂1 (𝑥 = 𝑑)) , 48 𝐵2′ = 𝐵𝑖 ′ (−𝜂1 (𝑥 = 𝑑)) , 𝐴3 = 𝐴𝑖(−𝜂2 (𝑥 = 𝑑)) , 𝐵3 = 𝐵𝑖(−𝜂2 (𝑥 = 𝑑)) , 𝐴′3 = 𝐴𝑖 ′ (−𝜂2 (𝑥 = 𝑑)), 𝐵3′ = 𝐵𝑖 ′ (−𝜂2 (𝑥 = 𝑑)), 𝐴𝑖′ and 𝐵𝑖 ′ are the first derivative of 𝐴𝑖 and 𝐵𝑖 with respect to 𝑖 2𝑚𝑒𝐹𝑑𝑖𝑒𝑙 1/3 their arguments, 𝜁 = 𝑘 ( ) , and 𝜉 = (𝐹/𝐹𝑑𝑖𝑒𝑙 )1/3. The transmission coefficient 𝑇3 is 0 ℏ2 2 𝑇3 = , (4.7) 𝜋[𝑃(𝑈 + 𝜁𝑌) − 𝜉𝑄(𝑉 + 𝜁𝑍)] where 𝑃 = 𝐴3 − 𝑖𝐵3 , 𝑄 = 𝐴′3 − 𝑖𝐵3′ , 𝑈 = 𝐴1 𝐵2′ − 𝐵1 𝐴′2 , 𝑉 = 𝐴1 𝐵2 − 𝐵1 𝐴2 , 𝑌 = 𝐴1′ 𝐵2′ − 𝐵1′ 𝐴′2 , and 𝑍 = 𝐴1′ 𝐵2 − 𝐵1′ 𝐴2. Note that the heterostructure in Fig. 4.1 can be easily solved numerically using transfer matrix approaches [106], [188]. Here, we expand such a set up analytically. The transmission probability, defined as 𝐷(𝜀) = 𝐽3 (𝜀)/𝐽𝑖 (𝜀), is the ratio of the transmitted probability current density in the vacuum to the incident probability current density in the metal, with probability current density 𝐽 = 𝑖ℏ/2𝑚(𝜓∇𝜓 ∗ − 𝜓 ∗ ∇𝜓), given by 1 1 2𝑚𝑒𝐹 1/3 4𝛼𝜉 1 𝐷(𝜀) = ( 2 ) |𝑇3 |2 = 3 2 , (4.8) 𝜋 𝑘0 ℏ 𝜋 Γ + Δ2 where Γ = 𝐴3 𝑈 + 𝛼𝐵3 𝑌 − 𝜉(𝐴′3 𝑉 + 𝛼𝐵3′ 𝑍), and Δ = 𝐵3 𝑈 − 𝛼𝐴3 𝑌 + 𝜉(𝛼𝐴′3 𝑍 − 𝐵3′ 𝑉), with 𝛼 = 1 2𝑚𝑒𝐹𝑑𝑖𝑒𝑙 1/3 |𝜁| = ( ) . 𝑘0 ℏ2 The electron emission current density can therefore be obtained by ∞ 𝐽 = 𝑒 ∫ 𝐷(𝜀)𝑁(𝜀)𝑑𝜀 , (4.9) 0 𝑚𝑘 𝑇 𝐸 −𝜀 where 𝐷(𝜀) is given in Eq. (4.8) and 𝑁(𝜀) = 2𝜋2𝐵ℏ3 ln [1 + exp ( 𝑘𝐹 𝑇 )] is the supply function as 𝐵 used in Eq. (2.9) of chapter 2, with 𝑘𝐵 the Boltzmann’s constant and 𝑇 the temperature. 4.3 Results and Discussion Figure 4.2 shows the tunneling probability for electrons with initial longitudinal energy of 𝜀 = 𝐸𝐹 , as a function of dielectric thickness 𝑑 and dielectric constant 𝜀𝑑𝑖𝑒𝑙 , under various combinations of dc electric field 𝐹 and dielectric electron affinity 𝜒, calculated from Eq. (4.8). The metal is assumed to be gold, with work function 𝑊 = 5.1 𝑒𝑉 and Fermi energy 𝐸𝐹 = 5.53 𝑒𝑉. Unless stated otherwise, these values are the default cathode metal properties in this chapter. When 𝐹 = 1 V/nm, as shown in Figs. 4.2(a) – 4.2(c), the electron tunneling probability 𝐷(𝐸𝐹 ) decreases with the dielectric constant 𝜀𝑑𝑖𝑒𝑙 for a given 𝑑. For 𝜀𝑑𝑖𝑒𝑙 > 2, 𝐷(𝐸𝐹 ) decreases as 𝑑 increases; however, for 𝜀𝑑𝑖𝑒𝑙 < 2, 𝐷(𝐸𝐹 ) increases as 𝑑 increases. As the dc electric field 49 increases to 𝐹 = 5 V/nm or 𝐹 = 10 V/nm, as shown in Figs. 4.2(d) – 4.2(i), it is obvious that the tunneling probability 𝐷(𝐸𝐹 ) increases, due to the narrowing of the surface potential barrier by the dc electric field. More importantly, there appear strong resonant peaks in 𝐷(𝐸𝐹 ). For a given 𝐹 and 𝜒, the resonant peaks shift to a larger value of 𝜀𝑑𝑖𝑒𝑙 as 𝑑 increases. The resonant peaks extend to a larger area in the 𝑑 − 𝜀𝑑𝑖𝑒𝑙 domain as either 𝐹 or 𝜒 increases. Figure 4.2 Electron tunneling probability 𝐷(𝜀 = 𝐸𝐹 ) as a function of the dielectric thickness 𝑑 and dielectric constant 𝜀𝑑𝑖𝑒𝑙 , under various combinations of dc electric field 𝐹 and the dielectric electron affinity 𝜒: (a) - (c) 𝐹 = 1 V/nm, (d) - (f) 𝐹 = 5 V/nm, (g) - (i) 𝐹 = 10 V/nm; with 𝜒 = 1 eV in (a), (d), (g); 𝜒 = 2 eV in (b), (e), (h); and 𝜒 = 3 eV in (c), (f), (i). The electron initial longitudinal energy is assumed to be Fermi energy 𝜀 = 𝐸𝐹 . The metal is assumed to be gold, with work function 𝑊 = 5.1 𝑒𝑉 and Fermi energy 𝐸𝐹 = 5.53 𝑒𝑉. The effects of dielectric thickness 𝑑, dielectric constant 𝜀𝑑𝑖𝑒𝑙 , and dielectric electron affinity 𝜒 on the electron tunneling probability 𝐷(𝐸𝐹 ) are further shown in line plots in Fig. 4.3. The solid curves, which give 𝐷(𝐸𝐹 ) as a function of 𝑑 in Figs. 4.3(a) – 4.3(c), show a parabola-like shape when 0 < 𝑑 < 𝑑0 , with a rough estimation of 𝑑0 (nm) ~ 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 . When 𝑑 > 𝑑0 , 𝐷(𝐸𝐹 ) oscillates around a constant and the oscillation amplitude decays with 𝑑. For a given set of 𝜀𝑑𝑖𝑒𝑙 and 𝜒 as in Fig. 4.3(a), when 𝐹 increases, the tunneling probability increases, due to the 50 narrowing of the potential barrier by the dc electric field. The resonance peaks shift towards smaller dielectric thickness values as 𝐹 increases, indicated by the dash line in Fig. 4.3(a). For fixed 𝐹 and 𝜒 as in Fig. 4.3(b), the tunneling probability decreases with increasing 𝜀𝑑𝑖𝑒𝑙 , due to the smaller dc electric field 𝐹𝑑𝑖𝑒𝑙 = 𝐹/𝜀𝑑𝑖𝑒𝑙 inside the dielectric. The dash line in Fig. 4.3(b) indicates that the resonance peaks shift to a large dielectric thickness value as 𝜀𝑑𝑖𝑒𝑙 increases, which is also consistent with the rough estimation of 𝑑0 (nm) ~ 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 . For a given combination of 𝐹 and 𝜀𝑑𝑖𝑒𝑙 as in Fig. 4.3(c), the tunneling probability increases with electron affinity 𝜒, due to the lowering of the potential barrier at metal-dielectric interface. It is shown that increasing 𝜒 makes the oscillation stronger; meanwhile, the resonance peaks slightly shift to a small dielectric thickness. It is interesting to find that the tunneling probability from coated cathodes can be larger than that from the uncoated ones. Some specific cases are highlighted by the yellow blocks in Figs. 4.3(b) and 4.3(c). For these cases, the dielectric-coated cathode has larger electron tunneling probability than the uncoated one, regardless of the thickness of the coating. Figures 4.3(b) and 4.3(c) show that the dielectric coating with small 𝜀𝑑𝑖𝑒𝑙 and large 𝜒 tends to enhance field emission from 1D flat cathode surfaces. Figures 4.3(d) – 4.3(f) show 𝐷(𝐸𝐹 ) as a function of 𝜀𝑑𝑖𝑒𝑙 for various 𝐹, 𝑑, and 𝜒. It is obvious that the tunneling probability decreases with dielectric constant, for a given combination of 𝐹, 𝑑 and 𝜒, due to the smaller dc electric field inside the dielectric. For fixed 𝐹 and 𝜒 in Fig. 4.3(e), as 𝑑 increases, there appear more resonance peaks for 𝐷(𝐸𝐹 ) as a function of 𝜀𝑑𝑖𝑒𝑙 . The curves for dielectric thickness 𝑑 > 2 nm start to overlap. That is because 𝐷(𝐸𝐹 ) gradually becomes constant against 𝑑 for a given combination of 𝜀𝑑𝑖𝑒𝑙 , 𝜒, and 𝐹, which has been shown in Figs. 4.3(a) – 4.3(c). Increasing 𝜒 results in more and stronger resonance peaks on curves for 𝐷(𝐸𝐹 ) vs 𝜀𝑑𝑖𝑒𝑙 , and extends those resonance peaks to large 𝜀𝑑𝑖𝑒𝑙 values, as shown in Fig. 4.3(f). 𝐷(𝐸𝐹 ) as a function of 𝜒 is shown in Figs. 4.3(g) – 4.3(i) for various 𝐹, 𝜀𝑑𝑖𝑒𝑙 , and 𝑑. The tunneling probability increases with 𝜒, for a given set of 𝐹, 𝜀𝑑𝑖𝑒𝑙 , and 𝑑, due to the lowering of the potential barrier at the metal-dielectric interface. When 𝜀𝑑𝑖𝑒𝑙 increases, 𝐷(𝐸𝐹 ) decreases, as shown in Fig. 4.3(h), due to the smaller dc electric field inside the dielectric and therefore wider potential barrier for an electron to tunnel through. When 𝐹 = 6 V/nm and 𝜀𝑑𝑖𝑒𝑙 = 4 in Fig. 4.3(i), the slope of the curves increases with 𝑑. It can be observed that the tunneling probability for cases of 𝑑 = 2 nm and 4 nm are larger than that for cases of 𝑑 = 0.5 nm and 1 nm for larger 𝜒, 51 which has also been shown in Figs. 4.3(a) – 4.3(c). Additionally, more resonance peaks are observed on curves for 𝐷(𝐸𝐹 ) as a function of 𝜒, when 𝑑 increases. Figure 4.3 Effects of dielectric thickness 𝑑, dielectric constant 𝜀𝑑𝑖𝑒𝑙 , dielectric electron affinity 𝜒, and dc electric field 𝐹 on the electron tunneling probability 𝐷(𝜀 = 𝐸𝐹 ) from dielectric-coated metal surface. 𝐷(𝜀 = 𝐸𝐹 ), calculated from Eq. (4.8), as a function of (a) – (c) dielectric thickness 𝑑; (d) – (f) dielectric constant 𝜀𝑑𝑖𝑒𝑙 ; and (g) – (i) dielectric electron affinity 𝜒. The electron initial energy is assumed to be Fermi level 𝐸𝐹 . The metal is assumed to be gold, with work function 𝑊 = 5.1 𝑒𝑉 and Fermi energy 𝐸𝐹 = 5.53 𝑒𝑉. Figure 4.4(a) shows the electron tunneling probability 𝐷(𝜀) as a function of electron initial longitudinal energy 𝜀 , for various combinations of dielectric coating thickness 𝑑 , dielectric constant 𝜀𝑑𝑖𝑒𝑙 and electron affinity 𝜒 , with the applied dc electric field 𝐹 = 7 V/nm. The tunneling probability increases as 𝜀 becomes larger, due to the lower potential barrier seen by electrons with larger 𝜀. Resonances appear at certain electron initial longitudinal energies, e.g., at 𝜀 ≈ 𝐸𝐹 , 7 𝑒𝑉 and 9 𝑒𝑉 for the case of 𝑑 = 1 nm, 𝜀𝑑𝑖𝑒𝑙 = 1.5, and 𝜒 = 2 𝑒𝑉 (orange curve in Fig. 4.4(a)). It is interesting to note that similar resonance behavior is also observed in previous 52 studies in electron tunneling through the metal-oxide-semiconductor structures [189], double- barrier semiconductors [190], metal surfaces with closely positioned positive ions [191], and nanostructured semiconductor film cathodes [192]. Figure 4.4(b) shows the electron emission current density per energy 𝐽(𝜀) = 𝑒𝑁(𝜀)𝐷(𝜀) as a function of 𝜀 for three temperatures at 𝑇 = 100, 300, and 2000 K. It is obvious that when 𝑇 = 2000 𝐾, more electrons with initial energies above the Fermi level are emitted, since more electrons are populated above the Fermi level at higher temperature. The contribution due to thermionic emission (i.e., emission of electrons with 𝜀 above the vacuum level) can be observed for 𝑇 = 2000 𝐾, as indicated in the red shaded area. The total emission current density (calculated from Eq. (4.9)) is 𝐽 = 3389, 3497, and 21150 A/cm2 at 𝑇 = 100, 300, and 2000 K, respectively. Figure 4.4 (a) Electron tunneling probability 𝐷(𝜀) as a function of electron initial longitudinal energy 𝜀, for various combinations of dielectric thickness 𝑑, dielectric constant 𝜀𝑑𝑖𝑒𝑙 and electron affinity 𝜒, at the applied dc field of 7 V/nm; (b) the emission current density per energy 𝐽(𝜀) = 𝑒𝑁(𝜀)𝐷(𝜀) under various temperatures 𝑇=100, 300, 2000 K, for fixed 𝑑 = 1 nm, 𝜀𝑑𝑖𝑒𝑙 = 1.5, 𝜒 = 1 eV, and 𝐹 = 7 V/nm. 53 Figure 4.5 Effects of dielectric thickness 𝑑, dielectric constant 𝜀𝑑𝑖𝑒𝑙 , dielectric electron affinity 𝜒, and dc electric field 𝐹 on the electron emission current density from dielectric-coated metal surfaces. The electron emission current density, calculated from Eq. (4.9), as a function of (a) – (c) dielectric thickness 𝑑; (d) – (f) dielectric constant 𝜀𝑑𝑖𝑒𝑙 ; and (g) – (i) dielectric electron affinity 𝜒. The metal is assumed to be gold, with work function 𝑊 = 5.1 𝑒𝑉 and Fermi energy 𝐸𝐹 = 5.53 𝑒𝑉. Figure 4.5 shows effects of dielectric thickness 𝑑, dielectric constant 𝜀𝑑𝑖𝑒𝑙 , electron affinity 𝜒, and dc electric field 𝐹, on the emission current density 𝐽 calculated from Eq. (4.9). Since most of the emitted electrons are with initial energies near Fermi level at room temperature, indicated in Fig. 4.4(b), the emission current density follows similar trends as the tunneling probability 𝐷(𝐸𝐹 ) in Fig. 4.3. The emission current density as a function of dielectric thickness, shows a parabola-like shape in the range of 0 < 𝑑 ≲ 𝑑0 [nm] = 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹. When 𝑑 ≳ 𝑑0 , 𝐽 oscillates around a constant. The behavior where the emission current density settles to a constant value for thick dielectric coatings under a fixed electric field is because the potential in the vacuum region drops below the electron initial energy, which does not contribute to the potential barrier for 54 electron tunneling. As a result, the emission current is determined by the barrier inside the dielectric region only and does not depend on the dielectric thickness, when the thickness is large. The oscillation can be ascribed to the interference between incident electron waves and reflected waves inside the dielectric layer [189]. It is found that resonance peaks on the curves for emission current density 𝐽 in Fig. 4.5 is not as sharp as those on electron tunneling probability 𝐷(𝐸𝐹 ) in Fig. 4.3. This can be explained by the emission of electrons with different initial energies, which, in combination, smooths the curve. The possible physical cause is the broadening of the resonances associated with the interference between reflected and transmitted waves for different initial energy of the electrons [188]. As 𝐹 increases, the emission current density increases, and resonance peaks shift towards smaller dielectric thickness 𝑑, as shown in Fig. 4.5(a). When the dielectric constant 𝜀𝑑𝑖𝑒𝑙 increases, the emission current density becomes smaller, due to the smaller dc electric field in the dielectric so that the barrier inside the dielectric is less narrowed. Meanwhile, an increased 𝜀𝑑𝑖𝑒𝑙 shifts the resonance peaks to a large thickness 𝑑, as shown in Fig. 4.5(b). When the electron affinity 𝜒 of the dielectric increases, the emission current density increases because of the lowering of the potential barrier at the metal-dielectric interface, and resonance peaks shift slightly to small thickness values. Similar to the tunneling probability in Fig. 4.3, a relatively small 𝜀𝑑𝑖𝑒𝑙 or a larger 𝜒 may induce a larger emission current density than the bare metal, as highlighted by the yellow blocks in Figs. 4.5(b) and 4.5(c). For the case of 𝐹 = 6 V/nm, 𝜀𝑑𝑖𝑒𝑙 = 2, and 𝜒 = 2 𝑒𝑉 in Fig. 4.5(c), 𝐽 is larger than that of the uncoated case only near a few resonant peaks, whereas 𝐷(𝐸𝐹 ) for this case in Fig. 4.3(c) is almost continuously larger than the uncoated case for 𝑑 > 1.5 nm. It can be explained by the fact that more electrons are emitted with initial longitudinal energies below Fermi level at room temperature, as shown in Fig. 4.4(b). For electrons with 𝜀 < 𝐸𝐹 , the tunneling probability from coated metals with dielectric thickness 𝑑 highlighted in Fig. 4.3(c) can be smaller than that from bare metal, thus yielding a smaller emission current density. The electron emission current density 𝐽 as a function of dielectric constant 𝜀𝑑𝑖𝑒𝑙 is shown in Figs. 4.5(d) – 4.5(f). When 𝜀𝑑𝑖𝑒𝑙 increases, 𝐽 decreases. As shown in Fig. 4.5(e), for the case of 𝐹 = 6 V/nm and 𝜒 = 2 𝑒𝑉, the curves for 𝑑 > 2 nm start to get overlapped, since the emission current density becomes almost constant as 𝑑 becomes large, as shown in Figs. 4.5(a) – 4.5(c). When 𝜒 increases in Fig. 4.5(f), more resonance peaks appear and extend towards larger value of 𝜀𝑑𝑖𝑒𝑙 . 55 Figures 4.5(g) – 4.5(i) show the electron emission current density as a function of dielectric electron affinity 𝜒. When 𝜒 increases, the emission current density increases. There is a sharp increase of the slope at 𝜒 ≈ 2.5 𝑒𝑉 for 𝐹 = 6 V/nm in Fig. 4.5(g). The sharp increases of the slope are also shown in Fig. 4.5(h) at 𝜒 ≈ 1.8 𝑒𝑉 and 𝜒 ≈ 2.8 𝑒𝑉 for the case of 𝜀𝑑𝑖𝑒𝑙 = 3, and at 𝜒 = 2.6 𝑒𝑉 for the case of 𝜀𝑑𝑖𝑒𝑙 = 4. These features are consistent with the resonant peaks of 𝐷(𝐸𝐹 ) observed in Figs. 4.3(g) and 4.3(h). When the dielectric thickness increases, the slope of the curves in Fig. 4.5(i) increases. At 𝜒 = 3 𝑒𝑉, the emission current density with 𝑑 = 2 nm and 4 nm exceeds that with 𝑑 = 0.5 nm and 1 nm in Fig. 4.5(i). This shows that at large 𝜒 where surfaces with thicker dielectric become better emitters, which is because a higher 𝜒 would lower the surface barrier, and a thicker dielectric layer would provide an overall smaller tunneling barrier (see, e.g., the barriers in Fig. 4.7). Figure 4.6 Emission current density as a function of dielectric thickness 𝑑 under various dielectric constants 𝜀𝑑𝑖𝑒𝑙 for (a) 𝜒 = 1 𝑒𝑉 and 𝐹 = 5 V/nm; (b) 𝜒 = 2 𝑒𝑉 and 𝐹 = 5 V/nm; and (c) 𝜒 = 1 𝑒𝑉 and 𝐹 = 10 V/nm. The metal is assumed to be gold, with work function 𝑊 = 5.1 𝑒𝑉 and Fermi energy 𝐸𝐹 = 5.53 𝑒𝑉. As already seen in Figs. 4.3 and 4.5, coating with a relatively small 𝜀𝑑𝑖𝑒𝑙 or a larger 𝜒 may induce a larger emission current density than the uncoated cathode. Figure 4.6 provides more calculations to determine the threshold values of dielectric thickness 𝑑𝑡ℎ and dielectric constant 𝑡ℎ 𝜀𝑑𝑖𝑒𝑙 , at which the emission current density 𝐽 is equal to that from the bare metal, for a given dielectric electron affinity 𝜒 and dc electric field 𝐹 . When 𝜒 = 1 eV and 𝐹 = 5 V/nm, the 𝑡ℎ thresholds are found to be 𝑑𝑡ℎ = 1.5 nm and 𝜀𝑑𝑖𝑒𝑙 = 1.38, as shown in Fig. 4.6(a). A dielectric 𝑡ℎ constant smaller than 𝜀𝑑𝑖𝑒𝑙 would enhance the electron emission compared to the uncoated case, 56 with thicknesses corresponding to the curves above the horizontal dash line. When the dielectric electron affinity increases to 𝜒 = 2 𝑒𝑉, the dielectric thickness and dielectric constant thresholds 𝑡ℎ becomes larger, i.e., 𝑑𝑡ℎ = 1.77 nm and 𝜀𝑑𝑖𝑒𝑙 = 2.09 in Fig. 4.6(b). When the dc field increases to 10 V/nm, the dielectric thickness and dielectric constant threshold shift towards smaller values, 𝑡ℎ i.e., 𝑑𝑡ℎ = 0.86 nm and 𝜀𝑑𝑖𝑒𝑙 = 1.35 in Fig. 4.6(c). Although it is difficult to give an exact expression to determine the dielectric constant and thickness thresholds, it is found that all three cases in Fig. 4.6 roughly follow the empirical relation at room temperature, 𝑡ℎ 𝜀𝑑𝑖𝑒𝑙 𝑊 𝑑𝑡ℎ [nm] = , (4.10) 𝑒𝐹 whose physical origin will become clear from the analysis of the potential barrier profiles in Fig. 4.7. Figures 4.7(a) – 4.7(c) show the potential profile induced by the dc electric field 𝐹 = 5 V/nm, for the coated metal with various dielectric thicknesses. The dielectric constants for Figs. 4.7(a) – 4.7(c) are 𝜀𝑑𝑖𝑒𝑙 = 1.2, 1.38, and 1.5, respectively, where 𝜀𝑑𝑖𝑒𝑙 = 1.38 is the dielectric constant threshold for the case of 𝜒 = 1 𝑒𝑉 and 𝐹 = 5 V/nm in Fig. 4.6(a). In the dielectric layer, the slope of the potential profile is 𝐹/𝜀𝑑𝑖𝑒𝑙 . To evaluate how the potential barrier affects the tunneling probability, the potential barrier width 𝑤(𝜀), indicated by the double-arrow line at initial energy 𝜀 in Fig. 4.7(a), and the corresponding tunneling probability 𝐷(𝜀) at 𝜀 = 𝐸𝐹 are plotted as a function of dielectric thickness 𝑑 in Figs. 4.7(d) – 4.7(f). The potential barrier width 𝑤(𝐸𝐹 ) increases with the dielectric thickness 𝑑 when 𝑑 < 𝜀𝑑𝑖𝑒𝑙 (𝑊 − 𝜒)/𝑒𝐹 (for arbitrary 𝜀, this is 𝜀𝑑𝑖𝑒𝑙 (𝑊 + 𝐸𝐹 − 𝜀 − 𝜒)/𝑒𝐹), due to the smaller electric field in the dielectric than that in the vacuum. 𝑤(𝐸𝐹 ) decreases in the dielectric thickness range of 𝜀𝑑𝑖𝑒𝑙 (𝑊 − 𝜒)/𝑒𝐹 < 𝑑 < 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹, where electrons need to tunnel through two separate barriers—one in the dielectric and the other in the vacuum. When 𝑑 > 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 (this is 𝜀𝑑𝑖𝑒𝑙 (𝑊 + 𝐸𝐹 − 𝜀)/𝑒𝐹 for arbitrary 𝜀), electrons with initial longitudinal energy of 𝐸𝐹 can be emitted by tunneling through only the barrier in the dielectric, and therefore the barrier width 𝑤(𝐸𝐹 ) reaches a constant for the same applied dc electric field. The corresponding electron tunneling probability, shown as blue curves in Figs. 4.7(d) – 4.7(f), decreases as 𝑤(𝐸𝐹 ) increases, and increases when 𝑤(𝐸𝐹 ) decreases. 𝐷(𝐸𝐹 ) reaches its first resonance peak at 𝑑 ≈ 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 , where electrons only need to tunnel through one potential barrier to emit. The red dash lines in Figs. 4.7(d) – 4.7(f) indicate that the maximum 57 resonance peak of 𝐷(𝐸𝐹 ) is roughly at the same thickness 𝑑 = 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 (cf. Eq. (4.10)), where 𝑤(𝐸𝐹 ) starts to become a constant. More calculations (not shown) for various combinations of 𝐹 and 𝜒 show that the maximum resonance peak of 𝐷(𝐸𝐹 ) can deviate but remain close to 𝑑 = 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹. The oscillation in 𝐷(𝐸𝐹 ) after the maximum is due to the interference between the incident waves and reflected wave at the dielectric-vacuum interface. It is also observed that when 𝜀𝑑𝑖𝑒𝑙 = 1.38, 𝐷(𝐸𝐹 ) by the coated metal with the dielectric thickness 𝑑 = 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 = 1.41 nm is larger than that by the bare metal, although potential barrier width of the coated case is larger than that of the bare one. By observing the potential profiles in Figs. 4.7(a) – 4.7(c), it is expected that the average constant value around which 𝐷(𝐸𝐹 ) oscillates at large 𝑑 may be estimated by, 4𝛼 1 𝐷(𝜀) = , (4.11) 𝜋 (𝐴1 + 𝛼𝐵1′ )2 + (𝛼𝐴1′ − 𝐵1 )2 with all the terms as defined in Sec. 4.2 [in Eqs. (4.6) and (4.8)], which is the electron tunneling probability due to a single triangular barrier (i.e., Fowler-Nordheim field emission [187], [193]) with potential barrier height 𝑊 − 𝜒 and electric field 𝐹𝑑𝑖𝑒𝑙 . Equation (4.11) is plotted as blue dotted lines in Figs. 4.7(d) – 4.7(f). 58 Figure 4.7 (a) – (c) Potential profile for coated metal with dielectric constants of (a) 𝜀𝑑𝑖𝑒𝑙 = 1.2; (b) 𝜀𝑑𝑖𝑒𝑙 = 1.38; (c) 𝜀𝑑𝑖𝑒𝑙 = 1.5 for various dielectric thicknesses 𝑑 (nm). (d) – (f) Barrier width 𝑤 seen by the electron with initial longitudinal energy of 𝜀 = 𝐸𝐹 , and the corresponding electron tunneling probability 𝐷 as a function of dielectric thickness 𝑑 for coatings with dielectric constants of (d) 𝜀𝑑𝑖𝑒𝑙 = 1.2; (e) 𝜀𝑑𝑖𝑒𝑙 = 1.38; (f) 𝜀𝑑𝑖𝑒𝑙 = 1.5. The applied dc electric field 𝐹 = 5 V/nm. Blue dotted lines are from Eq. (4.11). Thus, by comparing Eq. (4.11) and the tunneling probability from the uncoated cathode, one may determine the threshold value of dielectric constant 𝜀𝑑𝑖𝑒𝑙 , which is then used in Eq. (4.10) to give the threshold value of dielectric constant 𝑑𝑡ℎ , in order to have field emission current larger than the uncoated case from 1D flat surfaces. 4.4 Comparison with Modified Double-barrier Fowler-Nordheim Equation In this section, we compare our quantum model with the modified Fowler-Nordheim (FN) equation with a double-barrier potential profile, developed for cathode surfaces with an oxidation 59 layer [183], [184], 𝑒 3𝐹2 4√2𝑚 3 𝐽= 2 2 exp [− 𝑊 2 𝐶] (4.12) 16𝜋 ℏ𝑊𝐵 3𝑒ℏ𝐹 with 𝑊𝑒𝑓𝑓 𝑊𝑒𝑓𝑓 − 𝑒𝐹𝑑𝑖𝑒𝑙 𝑑 𝑊 − 𝑒𝐹𝑑𝑖𝑒𝑙 𝑑 𝐵 = 𝜀𝑑𝑖𝑒𝑙 [√ − 𝐻(𝑊𝑒𝑓𝑓 − 𝑒𝐹𝑑𝑖𝑒𝑙 𝑑)√ ] + 𝐻(𝑊 − 𝑒𝐹𝑑𝑖𝑒𝑙 𝑑)√ 𝑊 𝑊 𝑊 and 3 3 𝑊𝑒𝑓𝑓 2 𝑊𝑒𝑓𝑓 − 𝑒𝐹𝑑𝑖𝑒𝑙 𝑑 2 𝐶 = 𝜀𝑑𝑖𝑒𝑙 [( ) − 𝐻(𝑊𝑒𝑓𝑓 − 𝑒𝐹𝑑𝑖𝑒𝑙 𝑑) ( ) ] 𝑊 𝑊 3 𝑊 − 𝑒𝐹𝑑𝑖𝑒 𝑑 2 + 𝐻(𝑊 − 𝑒𝐹𝑑𝑖𝑒𝑙 𝑑) ( ) 𝑊 where 𝑒 and 𝑚 are the positive elementary charge and electron mass (set equal to electron rest mass in all the three regions of metal, dielectric, and vacuum), ℏ is the reduced Planck’s constant, 𝐹 is the applied dc electric field in the vacuum, 𝐹𝑑𝑖𝑒𝑙 = 𝐹/𝜀𝑑𝑖𝑒𝑙 is the electric field inside the dielectric layer, 𝑊 is the nominal work function of the metal, 𝑊𝑒𝑓𝑓 = 𝑊 − 𝜒 is the effective work function at metal-dielectric interface, and 𝑑 is the thickness of the dielectric layer, 𝐻(𝑥) is the Heaviside function. In case of no dielectric layer, 𝑊𝑒𝑓𝑓 = 𝑊 , 𝑑 = 0, 𝜀𝑑𝑖𝑒𝑙 = 1, 𝐵 and 𝐶 become 1, and Eq. (4.12) recovers the Fowler-Nordheim equation [97]. 60 Figure 4.8 (a) The emission current density 𝐽 from the dielectric-coated metal, calculated from Eq. (4.9), and the modified Fowler-Nordheim equation, Eq. (4.12), as a function of electric field under various combinations of dielectric thickness 𝑑, dielectric constant 𝜀𝑑𝑖𝑒𝑙 , and dielectric electron affinity 𝜒. (b) 𝐽 from the dielectric-coated metal, calculated from the quantum model (solid lines) and the modified Fowler-Nordheim equation (broken lines), as a function of dielectric thickness 𝑑. The dielectric has 𝜀𝑑𝑖𝑒𝑙 = 1.5, and 𝜒 = 2 𝑒𝑉. The electron emission current density of the quantum model, calculated from Eq. (4.9) is compared with that calculated from the modified FN equation, Eq. (4.12), as shown in Fig. 4.8. The two models show good agreement in the scaling, with the quantum model giving a higher emission current density in general. This is because the quantum model considers electron emission from all the energy levels at room temperature T = 300 K, whereas the FN based model assumes electron emission at 0 K. The quantum model shows resonance behavior in 𝐽 vs 𝑑, which cannot be revealed by the modified FN equation. 4.5 Concluding Remarks In this chapter, we have developed an exact theory for field emission from dielectric-coated cathode surfaces, by solving the one-dimensional time-independent Schrödinger equation with a double-triangular barrier, which is formed by applying dc electric field to the dielectric-coated cathode surface. The model includes both field emission and thermionic emission. It is found that the combination of a small dielectric constant and a large dielectric electron affinity tends to induce a larger emission current density than bare metal for 1D flat cathode surfaces, under a given dc electric field. It is found the emission current density is larger than the uncoated case 𝑡ℎ when the dielectric constant is smaller than a certain value 𝜀𝑑𝑖𝑒𝑙 and the dielectric thickness is 61 𝑡ℎ larger than the threshold value 𝑑𝑡ℎ [nm] ≈ 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 with 𝜀𝑑𝑖𝑒𝑙 < 𝜀𝑑𝑖𝑒𝑙 . This relation is consistent with the sharp transition of the potential barrier width in the double-triangular barrier. This quantum model is also compared with a modified Fowler-Nordheim equation for a double- triangular barrier, showing good agreement in the scaling of the emission current. The theory provides insights for designing the emitter with higher efficiency and better stability. 62 CHAPTER 5 PHOTOEMISSION FROM METAL SURFACES WITH NANOSCALE DIELECTRIC COATINGS AND ITS ENHANCEMENT BY PLASMONIC RESONANCE This chapter is based on the published journal papers “Theory of laser-induced photoemission from a metal surface with nanoscale dielectric coating,” J. Appl. Phys., vol. 131, no. 6, p. 064903, Feb. 2022, doi: 10.1063/5.0078060, by Yang Zhou and Peng Zhang, and “Plasmon-enhanced resonant photoemission using atomically thick dielectric coatings,” ACS Nano, vol. 14, no. 7, p. 8806-8815, Jun. 2020, doi: 10.1021/acsnano.0c03406, by X. Xiong, Yang Zhou, Y. Luo, X. Li, M. Bosman, L. K. Ang, Peng Zhang, and L. Wu. 5.1 Introduction High-performance photocathodes or photoemitters are always required to be of high efficiency and high stability [69], [106], [194]. Coatings, such as, graphene, nano-diamond, silicon dioxides, and zinc dioxides, are proposed to fabricated atop cathodes to protect them from degradation by ions and electrons bombardment, or oxidization under poor vacuum conditions [70]–[74]. Coatings not only elongate the operational lifetime and the current stability of photocathodes, but also enhance the quantum efficiency of photoemission by the lowering of the effective work function or the enhancement of laser field [47], [70], [71], [78], [79]. Analogous heterostructure photocathodes are prospective to optimize the quantum yield and emittance simultaneously for electron sources in X-ray free-electron lasers [69], [188]. The development of theory for photoemission from coated cathodes facilitates the optimization of the design and performance of photocathodes. Commonly used Fowler- Nordheim-type equations, which assume photoelectron emission occurs in positive half cycles of the intense laser, are applicable only in the optical field tunneling regime but not in the multiphoton absorption regime [184]. Furthermore, it has been shown that negative half cycles also play a role in the photoemission process [25]. Therefore, an exact model for photoemission from cathodes with ultrathin coatings is desirable to uncover the interplay of various parameters on photoemission and provide insights into the development of photocathodes. The introduction of surface coating results in an irregular double-barrier under the optical field. In Sec. 5.2, we approximate the double-barrier to an effective single triangular barrier by WKBJ approximation and employ the quantum photoemission model in Sec. 2.2, which is 63 developed by solving the TDSE subject to a triangular oscillating potential barrier, to investigate the photoemission processes under the plasmon resonant conditions on both bare and coated Au- nanopyramid field emitters. It is found that a 1 nm thick layer of SiO2 around a Au-nanopyramid will enhance photoemission current density by ~2 orders of magnitude and the optical field tunneling can be accessed at a significantly reduced incident laser intensity. In Sec. 5.3, we construct an exact analytical quantum model for laser-driven photoemission from cathodes coated with a nanoscale-thick dielectric by solving the TDSE. The model is applicable to photoemission for arbitrary combinations of metal properties (i.e., work function and Fermi level), dielectric properties (i.e., thickness, relative permittivity, and electron affinity), laser field (i.e., wavelength, and field strength or intensity), and dc field. Based on the analytical solution, we investigate the effects of dielectric properties on photoemission. This analytical model is compared with the effective single-triangular-barrier quantum model [47] and modified Fowler-Nordheim equation [183], [184] for photoemission from a dielectric-coated flat metal surface and a dielectric-coated pyramid-shaped nanoemitter. 5.2 Effective Single-triangular-barrier Quantum Model 5.2.1 Pyramid-shaped Nanoemitter Xiao Xiong and Lin Wu et al. [47] designed an efficient plasmonic photoemitter supporting antenna mode [195] by full-wave optical simulations [196]–[198], so that the incident optical energy could be maximally concentrated to the tip of the emitter. Figure 5.1(a) illustrates the schematics of the bare and coated Au-nanopyramid field emitters sitting on a Au substrate. The nanopyramid emitter has a side length of 𝑎 at the bottom surface or 𝜅𝑎 at the top surface and height ℎ, either bare in a vacuum or coated with a thin dielectric layer of thickness 𝑑 and refractive index 𝑛. The geometrical settings of the nanopyramid emitters (𝑎, 𝜅, and ℎ) are used to design the resonant wavelength of the antenna mode [47]. For a typical photoemitter with 𝑎 = ℎ = 40 nm and 𝜅 = 0.1, under the illumination of z- polarized light from the side, the antenna mode at 590 nm for the bare photoemitter or at 608 nm for the coated photoemitter (𝑑 = 1 nm, 𝑛 = 1.5) is observed, as shown in Fig. 1(a) in ref. [47]. Though occurring at similar resonant wavelengths, the plasmon field enhancement and the underlying microscopic physics differ drastically. For the bare Au-nanopyramid, a typical plasmonic nanostructure, the enhanced plasmonic field concentrates at the sharp corners of the Au with maximum field enhancement 𝛽𝐴𝑢 = 35, and its magnitude exponentially decays into the 64 vacuum, dying out at a distance of <10 nm. In contrast, for the coated case, the dielectric coating with a refractive index 𝑛 creates an interface with the vacuum. This interface helps to reflect and confine the plasmonic fields into an even smaller volume [195], effectively forming a dielectric waveguide that can locally enhance the fields at the Au surface [199]. As a consequence, the maximum field enhancement at the Au surface 𝛽𝐴𝑢 increases from 35 to 200 (corresponding to an optical intensity enhancement from 1225 to 40000) due to the combined effects of geometrical plasmon field enhancement and a secondary field enhancement from the plasmonic field confinement. The tremendously increased field enhancement 𝛽𝐴𝑢 at the Au surface from the antenna mode is particularly desirable, which can vastly facilitate the photoemission process to operate at the optical field tunneling regime at a much lower incident field strength. Figure 5.1 (a) Left: Geometrical settings of Au-nanopyramid field emitter; and Right: Schematics of the bare and coated Au-nanopyramid field emitter sitting on a Au substrate, illuminated by z-polarized light from the side. (b) Potential profile of the tunneling barriers induced by different incident laser field strengths F for bare (top) and coated (bottom) Au- nanopyramid field emitter. 5.2.2 Effective Field Enhancement To understand the photoemission process, we start with the time-dependent potential barrier 𝜙(𝑧, 𝑡) that is faced by the free electrons in Au to tunnel through and emit. In the dielectric 65 region, 0 < 𝑧 < 𝑑, it is written as: 𝑧 𝛽𝐷 − 𝛽𝐴𝑢 2 𝜙(𝑧, 𝑡) = 𝑉0 − 𝑒𝐹 cos 𝜔𝑡 ∫ 𝛽(𝑠)𝑑𝑠 ≅ 𝑉0 − 𝑒𝐹 cos 𝜔𝑡 ( 𝑧 + 𝛽𝐴𝑢 𝑧) (5.1) 0 2𝑑 Here, 𝑧 denotes the distance to the top surface of the Au tip; 𝑉0 = 𝑊 + 𝐸𝐹 − 𝜒 is the nominal potential barrier height at the Au surface, where 𝑊 = 5.1 eV and 𝐸𝐹 = 5.53 eV are the work function and Fermi energy of Au and 𝜒 is the electron affinity of the dielectric layer; 𝑒 is the elementary charge (positive); 𝜔 = 2𝜋𝑐/𝜆 denotes the angular frequency with the laser wavelength 𝜆, where 𝑐 is the speed of light in vacuum; and 𝐹 represents the field strength of the incident laser. The optical near fields are taken care of by the exact enhancement profile that is extracted from optical simulations [47] and fitted using a linear function of distance, 𝛽(𝑧) = 𝛽𝐴𝑢 − (𝛽𝐴𝑢 − 𝛽𝐷 )𝑧/𝑑 , with 𝛽(0) = 𝛽𝐴𝑢 at the Au surface and 𝛽(𝑑) = 𝛽𝐷 at the dielectric- vacuum interface, respectively. In the free space region, 𝑧 > 𝑑, the potential profile reads, 𝛽𝐴𝑢 − 𝛽𝐷 𝜙(𝑧, 𝑡) = 𝑊 + 𝐸𝐹 − 𝑒𝐹 cos 𝜔𝑡 ( 𝑑 + 𝛽𝐷 𝑧) (5.2) 2 where the field enhancement is assumed constant 𝛽𝐷 . To ensure a fair comparison, our optical simulations employ exactly the same settings for both bare and coated field emitters, but we set 𝑛 = 1 to the dielectric for the bare emitter. As a result, the potential profiles described above also apply to the bare emitter, where we set the electron affinity 𝜒 = 0 for the dielectric. We plot in Figure 5.1(b) their potential barrier profiles 𝜙(𝑧, 𝑡 = 0) induced by a different incident laser. Clearly, the presence of a dielectric coating not only reduces the height of the potential barrier due to the electron affinity of the dielectric layer, 𝜒, but also significantly narrows the barrier, because of the much stronger field enhancement at the Au surface, 𝛽𝐴𝑢 . This barrier narrowing effect becomes even more profound for larger incident laser fields, as indicated by blue dashed lines in Fig. 5.1(b). Despite the barrier narrowing effect from the dielectric coatings, electrons inside the metal now need to overcome two barriers to get photoemitted (Figure 5.1(b)), for example via multiphoton absorption, photo-assisted tunneling through either vacuum or the dielectric layer, or direct optical field tunneling. The probability for each of these processes depends on the electron initial energy 𝜀 and the overall potential barrier for a given incident laser field 𝐹. Here, 𝜀 is the longitudinal energy of electrons inside the metal impinging on the metal surface. To calculate the photoemission current, we employ the quantum theory developed in Sec. 2.2. 66 Therefore, we first need to approximate the irregular “double-barrier” potential profiles with an effective triangular barrier, as exemplified in Fig. 5.2(a). Here, the effective work function of the coated Au is fixed at 𝑊𝑒𝑓𝑓 = 𝑊 − 𝜒. For an electron inside the metal with an initial energy 𝜀, it would originally see an irregular potential barrier at the metal surface (left of Fig. 5.2(a)). Correspondingly, the effective triangular barrier (right of Fig. 5.2(a)) has the same barrier height as the original barrier, 𝑊𝑒𝑓𝑓 + 𝐸𝐹 − 𝜀, for electrons at energy level 𝜀, but with potential energy dropping linearly as a function of the distance away from the metal surface 𝑧. The area under the two barriers is kept the same (cf. the red-shaded area 𝑆 vs the blue-shaded area 𝑆′ in Fig. 5.2(a)), from which the effective field enhancement factor 𝛽𝑒𝑓𝑓 (𝜀) can be determined for electrons with initial energy 𝜀. As the area under the irregular double barrier changes nonlinearly with both the electron initial energy 𝜀 and the field strength of the incident laser 𝐹 , the effective field enhancement factor 𝛽𝑒𝑓𝑓 is also a function of 𝜀 and 𝐹 . This effective triangular-barrier approximation is plausible as the electron transmission is insensitive to the actual shape of the barrier, but is predominantly determined by the “area under the curve” according to the WKBJ approximation [106], [200], which has been previously verified for photoemission [38], [49], [62], [100], [119]. We also compare this effective single-triangular-barrier quantum photoemission model based on this approximation to a double-barrier Fowler−Nordheim equation used in the “simple-man” model [183], [184] and exact analytical quantum model in Sec. 5.4. In Fig. 5.2(b), we plot the 𝜀-dependent effective optical field enhancement factor 𝛽𝑒𝑓𝑓 as a function of the field strength of the incident laser 𝐹 at three representative electron initial energies 𝜀 for both coated and bare photoemitters. The coated emitter has improved 𝛽𝑒𝑓𝑓 over the bare emitter only for 𝐹 exceeding a certain threshold, e.g., 0.018 V/nm for Fermi electrons with 𝜀 = 𝐸𝐹 . This can be explained by the barrier profiles in Fig. 5.1(b). When 𝐹 is small (black solid lines), the second barrier peak at the dielectric-vacuum interface can be higher than that at the Au surface. This results in 𝛽𝑒𝑓𝑓 smaller than that in the bare emitter due to the double-barrier profile. But when 𝐹 is medium to large (blue dashed lines), the second barrier peak is lowered, and 𝛽𝑒𝑓𝑓 is predominantly determined by the significantly increased field enhancement at the Au surface 𝛽𝐴𝑢 . 67 Figure 5.2 (a) Triangular-barrier approximation: with fixed effective work function 𝑊𝑒𝑓𝑓 = 𝑊 − 𝜒, an effective field enhancement 𝛽𝑒𝑓𝑓 is defined for each electron initial energy 𝜀 to maintain the same area under the barriers: 𝑆 ′ = 𝑆. (b) Calculated 𝛽𝑒𝑓𝑓 for three representative 𝜀. 5.2.3 Results and Discussion The calculated emission current density 𝐽 as a function of the field strength of the incident laser 𝐹 is shown in Fig. 5.3(a). In regime I where 𝐹 < 0.014 V/nm, the emission current density from the coated Au tip is close to that from the bare tip, because of the smaller effective enhancement factor 𝛽𝑒𝑓𝑓 , as indicated by the lowered but widened barrier for Fermi electrons at 𝐹 = 0.01 V/nm (inset). However, in regime II where 𝐹 > 0.014 V/nm, the coated Au photoemitter outperforms the bare photoemitter, due to the combined effects of increased 𝛽𝑒𝑓𝑓 and lowered potential barrier, e.g., for Fermi electrons at 𝐹 = 0.1 V/nm (inset). For laser fields over a wide range of 𝐹 = 0.014 − 0.1 V/nm, 𝐽 from the coated photoemitter is enhanced by at least 2 orders of magnitude as compared to the bare emitter. The threshold laser field of 0.014 V/nm, at which the coated emitter outperforms the bare emitter, can also be derived when 𝛽𝑒𝑓𝑓 of the coated emitter exceeds that of the bare emitter for Fermi electrons, i.e., 0.018 V/nm in Fig. 5.2(b). Interestingly, the increment of the current density 𝐽 from the coated emitter slows down at higher incident laser fields 𝐹 > 0.05 V/nm (Fig. 5.3(a)), due to the saturation of 𝛽𝑒𝑓𝑓 at larger 𝐹 (Fig. 5.2(b)). The decreased slope suggests that the coated emitter has probably entered into the optical field tunneling regime, as it follows the Fowler−Nordheim current density scaling law [45], [119]. 68 Figure 5.3 (a) Calculated photoemission current density 𝐽 as a function of the field strength of the incident laser. (b) Photoelectron energy spectra via j-photon absorption (with respect to 𝐸𝐹 ) at 𝐹 = 0.01 or 0.1 V/nm. In all calculations, the dielectric coating has 𝑑 = 1 nm, 𝑛 = 1.5, and 𝜒 = 0.9 eV. To better understand the photoemission mechanism, we plot the photoelectron energy spectra for two representative incident laser fields in Fig. 5.3(b). At 𝐹 = 0.01 V/nm, the spectra exhibit distinct multiphoton peaks, whose magnitudes decay rapidly with energy. The dominant peaks correspond to three-photon absorption, as the ratio of barrier height to photon energy 𝑊𝑒𝑓𝑓 /ℏ𝜔 > 2 for both bare and coated photoemitters. At larger field 𝐹 = 0.1 V/nm, the spectrum of the coated emitter becomes significantly broadened, reaching a plateau; meanwhile, the multiphoton peaks are severely smeared out. The broadening of the spectrum is ascribable to the increased contributions of higher-order photon processes at large incident laser fields [38], as electrons need to absorb sufficient photon energy to overcome the increased ponderomotive 2 energy, 𝑈𝑝 = 𝑒 2 (𝛽𝑒𝑓𝑓 𝐹) /4𝑚𝜔2 , in order to emit. Classically, the plateau also signifies the back‑propagation and rescatterings of electrons in an optical field tunneling process [11], [49], [184]. Therefore, the features of broadening and the plateau in the energy spectrum for the coated photoemitter indicate a more rapid transition from multiphoton absorption to optical field tunneling [24], [38], [49]. To be quantitative, the transition point can be indicatively determined by a local Keldysh parameter at the tip 𝛾𝑙𝑜𝑐 = 𝜔√2𝑚𝑊𝑒𝑓𝑓 /𝑒𝛽𝑒𝑓𝑓 𝐹 ≈ 1 [24], [38], [49], [119], [120], which gives a corresponding transition incident field strength of 0.10 (or 0.32) V/ nm for the coated (or bare) field emitter. Alternatively, the transition point could be nominally marked by observing the termination of the scaling 𝐽 ∝ 𝐹 2𝑗 with 𝑗 = 3, which indicates the exit of the three-photon absorption regime, yielding a transition incident field strength of 0.05 (or 0.17) V/nm for the coated (or bare) field emitter. In either way, the coated emitter reaches the optical field tunneling regime at less than one-third of the incident laser field 𝐹 as compared to the bare 69 emitter. In other words, the optical field tunneling regime can be accessed at an incident laser intensity of about 10 times smaller with the dielectric coating. 5.3 1D Exact Analytical Quantum Model 5.3.1 Formulation Figure 5.4 Photoemission from a flat metal surface coated with a dielectric under a laser electric field and a dc bias. The metal-dielectric interface is located at 𝑥 = 0, and the coating’s thickness is 𝑑. The metal has Fermi energy 𝐸𝐹 and nominal work function of 𝑊0 . The effective work function 𝑊 = 𝑊0 − Δ𝑊 , with the Schottky barrier lowering Δ𝑊 = 2√𝑒 3 𝐹0𝑑𝑖𝑒𝑙 /16𝜋𝜀0 𝜀𝑑𝑖𝑒𝑙 when the maximum of the potential barrier including image charge potential is in the coating or Δ𝑊 = 2√𝑒 3 𝐹0 /16𝜋𝜀0 when the potential maximum is in the vacuum. The dielectric has an electron affinity of 𝜒 and a relative permittivity of 𝜀𝑑𝑖𝑒𝑙 . The laser field strengths are 𝐹1 in the vacuum and 𝐹1𝑑𝑖𝑒𝑙 in the coating. The dc field strengths are 𝐹0 in the vacuum and 𝐹0𝑑𝑖𝑒𝑙 in the coating. The electron incident longitudinal energy is 𝜀 . The black solid line represents the potential profile under the dc field 𝐹0 only, and the red dotted lines are for the time-dependent potential profile due to both 𝐹0 and 𝐹1 . Slopes of the potential profile, denoted as 𝑆1, 𝑆2 , 𝑆3 , and 𝑆4 , are −𝑒𝐹0𝑑𝑖𝑒𝑙 , −𝑒𝐹0 , −𝑒(𝐹0𝑑𝑖𝑒𝑙 + 𝐹1𝑑𝑖𝑒𝑙 ), and −𝑒(𝐹0 + 𝐹1 ), respectively. In the 1D model (see Fig. 5.4), electrons with initial longitudinal energy 𝜀 are emitted from the flat metal surface coated with a nanoscale-thick dielectric, driven by a laser field and a dc bias. The laser field and dc bias field are perpendicular to the metal surface. For simplicity, the scattering effects of photoexcited electrons with other electrons and phonons in the metal and dielectric, the charge trapping effect in the dielectric, are ignored [25], [38], [125]. Therefore, the time-varying potential barrier in those three regions, i.e., metal, dielectric, and vacuum, reads, 70 0, 𝑥<0 𝑑𝑖𝑒𝑙 𝑑𝑖𝑒𝑙 𝜙(𝑥, 𝑡) = { 𝑉0 − 𝜒 − 𝑒𝐹0 𝑥 − 𝑒𝐹1 𝑥 cos 𝜔𝑡 , 0≤𝑥<𝑑 (5.3) 𝑑𝑖𝑒𝑙 𝑑𝑖𝑒𝑙 𝑉0 + 𝑒𝑑(𝐹0 − 𝐹0 ) + 𝑒𝑑(𝐹1 − 𝐹1 ) cos 𝜔𝑡 − 𝑒𝐹0 𝑥 − 𝑒𝐹1 𝑥 cos 𝜔𝑡 , 𝑥 ≥ 𝑑 where 𝑉0 = 𝑊 + 𝐸𝐹 , with 𝐸𝐹 the Fermi energy of the metal and 𝑊 = 𝑊0 − Δ𝑊 the effective work function including the potential barrier lowering by the Schottky effect due to the dc electric field 𝐹0 , Δ𝑊 = 2√𝑒 3 𝐹0𝑑𝑖𝑒𝑙 /16𝜋𝜀0 𝜀𝑑𝑖𝑒𝑙 when the maximum of the potential barrier including image charge potential is in the coating or Δ𝑊 = 2√𝑒 3 𝐹0 /16𝜋𝜀0 when the potential maximum is in the vacuum; 𝜒 is the electron affinity of the dielectric; 𝑒 is the positive elementary charge; 𝐹0 and 𝐹1 are the dc electric field and laser electric field in the vacuum respectively; 𝐹0𝑑𝑖𝑒𝑙 and 𝐹1𝑑𝑖𝑒𝑙 are the dc and laser electric field inside the dielectric respectively; 𝜔 is the angular frequency of the laser field; and 𝑑 is the thickness of the dielectric. For perfectly flat surface, 𝐹0𝑑𝑖𝑒𝑙 = 𝐹0 /𝜀𝑑𝑖𝑒𝑙 and 𝐹1𝑑𝑖𝑒𝑙 = 𝐹1 /𝜀𝑑𝑖𝑒𝑙 inside the dielectric with 𝜀𝑑𝑖𝑒𝑙 the relative permittivity of the dielectric. The electron wave functions 𝜓(𝑥, 𝑡) in the metal, dielectric, and vacuum are obtained by solving the time-dependent Schrödinger equation, 𝜕𝜓(𝑥, 𝑡) ℏ2 𝜕 2 𝜓(𝑥, 𝑡) 𝑖ℏ =− + 𝜙(𝑥, 𝑡)𝜓(𝑥, 𝑡) (5.4) 𝜕𝑡 2𝑚 𝜕𝑥 2 where ℏ is the reduced Planck’s constant; 𝑚 is the electron effective mass, set to the electron rest mass in all three regions for simplicity; and 𝜙(𝑥, 𝑡) is the potential given in Eq. (5.3). The exact solution to Eq. (5.4) in the metal (𝑥 < 0) is ∞ 𝜀 𝜀+𝑛ℏ𝜔 𝜓𝐼 (𝑥, 𝑡) = 𝑒 −𝑖ℏ𝑡+𝑖𝑘0 𝑥 + ∑ 𝑅1𝑛 𝑒 −𝑖 ℏ 𝑡−𝑖𝑘𝑛 𝑥 , 𝑥<0 (5.5) 𝑛=−∞ where 𝑘0 = √2𝑚𝜀/ℏ2 and 𝑘𝑛 = √2𝑚(𝜀 + 𝑛ℏ𝜔)/ℏ2 . It represents the superposition of the incident plane wave with electron incident longitudinal energy 𝜀, and a set of reflected waves with energy of 𝜀 + 𝑛ℏ𝜔 after photon absorption (𝑛 > 0) or emission (𝑛 < 0) processes. 𝑅1𝑛 is the reflection coefficient. In the dielectric (0 ≤ 𝑥 < 𝑑), the solution to Eq. (5.4) is obtained by following Truscott transformation and separation of variables [25], [38], [91]–[93], [121], [125], 71 ∞ 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 cos 𝜔𝑡 𝜓𝐼𝐼 (𝑥, 𝑡) = ∑ 𝑇2𝑛 exp [𝑖 √ 𝐸 (𝑥 + )] Θ(𝑥, 𝑡) ℏ2 2𝑛 𝑚𝜔 2 𝑛=−∞ ∞ 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 cos 𝜔𝑡 + ∑ 𝑅2𝑛 exp [−𝑖√ 𝐸 (𝑥 + )] Θ(𝑥, 𝑡) , 0 ≤ 𝑥 < 𝑑 ℏ2 2𝑛 𝑚𝜔 2 𝑛=−∞ (5.6𝑎) for 𝐹0 = 0, or ∞ 𝜓𝐼𝐼 (𝑥, 𝑡) = ∑ [𝑇2𝑛 [𝐴𝑖(−𝜁𝑛 ) − 𝑖𝐵𝑖(−𝜁𝑛 )] + 𝑅2𝑛 [𝐴𝑖(−𝜁𝑛 ) + 𝑖𝐵𝑖(−𝜁𝑛 )]]Γ(𝑥, 𝑡) , 0 ≤ 𝑥 < 𝑑 𝑛=−∞ (5.6𝑏) for 𝐹0 ≠ 0, where 𝑇2𝑛 and 𝑅2𝑛 are the transmission coefficient and reflection coefficient of electron waves through the nth channel in the dielectric, respectively; 𝐸2𝑛 = 𝜀 + 𝑛ℏ𝜔 − 𝑈𝑝2 − 2 𝑒 2 (𝐹1𝑑𝑖𝑒𝑙 ) 𝑉20 is the drift kinetic energy in the dielectric, with the ponderomotive energy 𝑈𝑝2 = 4𝑚𝜔 2 2 𝜀+𝑛ℏ𝜔 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑒 2 (𝐹1𝑑𝑖𝑒𝑙 ) sin 2𝜔𝑡 and 𝑉20 = 𝐸𝐹 + 𝑊 − 𝜒 ; Θ(𝑥, 𝑡) = exp [−𝑖 𝑡+𝑖 𝑥+𝑖 ] ; ℏ ℏ𝜔 8ℏ𝑚𝜔 3 1/3 𝑒 2 𝐹0𝑑𝑖𝑒𝑙 𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑒𝐹1𝑑𝑖𝑒𝑙 cos 𝜔𝑡 𝐸2𝑛 2𝑚𝑒𝐹0𝑑𝑖𝑒𝑙 Γ(𝑥, 𝑡) = exp [−𝑖 ] Θ(𝑥, 𝑡) ; 𝜁𝑛 = (𝑥 + + 𝑒𝐹𝑑𝑖𝑒𝑙 ) ( ) . 𝜓𝐼𝐼 ℏ𝑚𝜔 3 𝑚𝜔 2 0 ℏ2 represents the superposition of the forward traveling waves and the reflected waves in the dielectric. In the vacuum (𝑥 ≥ 𝑑), the exact solution to Eq. (5.4) is ∞ 2𝑚 𝑒𝐹1 cos 𝜔𝑡 𝜓𝐼𝐼𝐼 (𝑥, 𝑡) = ∑ 𝑇3𝑛 exp [𝑖√ 2 𝐸3𝑛 (𝑥 + )] Ξ(𝑥, 𝑡) , 𝑥 ≥ 𝑑 (5.7𝑎) ℏ 𝑚𝜔 2 𝑛=−∞ for 𝐹0 = 0, or ∞ 𝑒 2 𝐹0 𝐹1 sin 𝜔𝑡 𝜓𝐼𝐼𝐼 (𝑥, 𝑡) = ∑ 𝑇3𝑛 [𝐴𝑖(−𝜂𝑛 ) − 𝑖𝐵𝑖(−𝜂𝑛 )] exp [−𝑖 ] Ξ(𝑥, 𝑡), 𝑥 ≥ 𝑑 (5.7𝑏) ℏ𝑚𝜔 3 𝑛=−∞ for 𝐹0 ≠ 0, where 𝑇3𝑛 is the transmission coefficient in the vacuum; 𝐸3𝑛 = 𝜀 + 𝑛ℏ𝜔 − 𝑈𝑝3 − 𝑒 2 𝐹2 𝑉30 is the drift kinetic energy in the vacuum, with the ponderomotive energy 𝑈𝑝3 = 4𝑚𝜔12 and 𝑉30 = 𝑊 + 𝐸𝐹 + 𝑒𝑑(𝐹0 − 𝐹0𝑑𝑖𝑒𝑙 ) for 𝐹0 ≠ 0 or 𝑉30 = 𝑊 + 𝐸𝐹 for 𝐹0 = 0 ; Ξ(𝑥, 𝑡) = 72 𝑒𝑑(𝐹1 −𝐹1𝑑𝑖𝑒𝑙 ) sin 𝜔𝑡 𝜀+𝑛ℏ𝜔 𝑒𝐹1 sin 𝜔𝑡 𝑒 2 𝐹12 sin 2𝜔𝑡 𝑒𝐹1 cos 𝜔𝑡 exp [−𝑖 −𝑖 𝑡+𝑖 𝑥+𝑖 ] ; 𝜂𝑛 = (𝑥 + + ℏ𝜔 ℏ ℏ𝜔 8ℏ𝑚𝜔 3 𝑚𝜔2 𝐸3𝑛 2𝑚𝑒𝐹 1/3 𝑒𝐹0 ) ( ℏ2 0 ) . 𝜓𝐼𝐼𝐼 denotes the outgoing waves traveling to the vacuum. Continuity of the wave function and its derivative at both metal-dielectric interface (𝑥 = 0) and dielectric-vacuum interface (𝑥 = 𝑑) and Fourier transform yield the solution for 𝑅1𝑛 , 𝑇2𝑛 , 𝑅2𝑛 , and 𝑇3𝑛 , ∞ ∞ 𝛿(𝑙) + 𝑅1𝑙 = ∑ 𝑇2𝑛 𝑃1𝑛(𝑛−𝑙) + ∑ 𝑅2𝑛 𝑄1𝑛(𝑛−𝑙) (5.8𝑎) 𝑛=−∞ 𝑛=−∞ ∞ ∞ 𝑘0 𝛿(𝑙) − 𝑘𝑙 𝑅1𝑙 = ∑ 𝑇2𝑛 𝑃2𝑛(𝑛−𝑙) + ∑ 𝑅2𝑛 𝑄2𝑛(𝑛−𝑙) (5.8𝑏) 𝑛=−∞ 𝑛=−∞ ∞ ∞ ∞ ∑ 𝑇2𝑛 𝑃3𝑛(𝑛−𝑙) + ∑ 𝑅2𝑛 𝑄3𝑛(𝑛−𝑙) = ∑ 𝑇3𝑛 𝑍3𝑛(𝑛−𝑙) (5.8𝑐) 𝑛=−∞ 𝑛=−∞ 𝑛=−∞ ∞ ∞ ∞ ∑ 𝑇2𝑛 𝑃4𝑛(𝑛−𝑙) + ∑ 𝑅2𝑛 𝑄4𝑛(𝑛−𝑙) = ∑ 𝑇3𝑛 𝑍4𝑛(𝑛−𝑙) (5.8𝑑) 𝑛=−∞ 𝑛=−∞ 𝑛=−∞ 1 2𝜋 where 𝛿 is the Dirac delta function; 𝑃1𝑛𝑙 = 2𝜋 ∫0 𝑃1𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) , 𝑄1𝑛𝑙 = 1 2𝜋 1 2𝜋 ∫ 𝑄1𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) 2𝜋 0 , 𝑃2𝑛𝑙 = 2𝜋 ∫0 𝑃2𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) , 𝑄2𝑛𝑙 = 1 2𝜋 1 2𝜋 ∫ 𝑄2𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) 2𝜋 0 , 𝑃3𝑛𝑙 = 2𝜋 ∫0 𝑃3𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) , 𝑄3𝑛𝑙 = 1 2𝜋 1 2𝜋 ∫ 𝑄3𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) 2𝜋 0 , 𝑍3𝑛𝑙 = 2𝜋 ∫0 𝑍3𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) , 𝑃4𝑛𝑙 = 1 2𝜋 1 2𝜋 ∫ 𝑃4𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) 2𝜋 0 , 𝑄4𝑛𝑙 = 2𝜋 ∫0 𝑄4𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) , 𝑍4𝑛𝑙 = 1 2𝜋 ∫ 𝑍4𝑛 (𝜔𝑡)𝑒 −𝑖𝑙𝜔𝑡 𝑑(𝜔𝑡) 2𝜋 0 are the Fourier transform coefficients. For 𝐹0 = 0, we have 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 cos 𝜔𝑡 𝑃1𝑛 (𝜔𝑡) = exp [𝑖 √ 𝐸 ( )] 𝜌(𝜔𝑡) (5.9𝑎) ℏ2 2𝑛 𝑚𝜔 2 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 cos 𝜔𝑡 𝑄1𝑛 (𝜔𝑡) = exp [−𝑖 √ 𝐸 ( )] 𝜌(𝜔𝑡) (5.9𝑏) ℏ2 2𝑛 𝑚𝜔 2 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑃2𝑛 (𝜔𝑡) = 𝑃1𝑛 (𝜔𝑡) [√ 𝐸 + ] (5.9𝑐) ℏ2 2𝑛 ℏ𝜔 73 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑄2𝑛 (𝜔𝑡) = 𝑄1𝑛 (𝜔𝑡) [−√ 𝐸 + ] (5.9𝑑) ℏ2 2𝑛 ℏ𝜔 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 cos 𝜔𝑡 𝑃3𝑛 (𝜔𝑡) = exp [𝑖 √ 𝐸 (𝑑 + )] 𝜌(𝜔𝑡) (5.9𝑒) ℏ2 2𝑛 𝑚𝜔 2 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 cos 𝜔𝑡 𝑄3𝑛 (𝜔𝑡) = exp [−𝑖 √ 𝐸 (𝑑 + )] 𝜌(𝜔𝑡) (5.9𝑓) ℏ2 2𝑛 𝑚𝜔 2 2𝑚 𝑒𝐹1 cos 𝜔𝑡 𝑍3𝑛 (𝜔𝑡) = exp [𝑖 √ 𝐸 3𝑛 (𝑑 + )] 𝜚(𝜔𝑡) (5.9𝑔) ℏ2 𝑚𝜔 2 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑃4𝑛 (𝜔𝑡) = 𝑃3𝑛 (𝜔𝑡) [√ 2 𝐸2𝑛 + ] (5.9ℎ) ℏ ℏ𝜔 2𝑚 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑄4𝑛 (𝜔𝑡) = 𝑄3𝑛 (𝜔𝑡) [−√ 𝐸 + ] (5.9𝑖) ℏ2 2𝑛 ℏ𝜔 2𝑚 𝑒𝐹1 sin 𝜔𝑡 𝑍4𝑛 (𝜔𝑡) = 𝑍3𝑛 (𝜔𝑡) [√ 𝐸3𝑛 + ]. (5.9𝑗) ℏ2 ℏ𝜔 2 𝑒 2 (𝐹1𝑑𝑖𝑒𝑙 ) sin 2𝜔𝑡 𝑒 2 𝐹12 sin 2𝜔𝑡 with 𝜌(𝜔𝑡) = exp [𝑖 ] and 𝜚(𝜔𝑡) = exp [𝑖 ]. 8ℏ𝑚𝜔 3 8ℏ𝑚𝜔 3 For 𝐹0 ≠ 0, we have, 𝑃1𝑛 (𝜔𝑡) = 𝑠(𝜁𝑛 (𝑥 = 0))Υ(𝜔𝑡) (5.10𝑎) 𝑄1𝑛 (𝜔𝑡) = 𝑟(𝜁𝑛 (𝑥 = 0))Υ(𝜔𝑡) (5.10𝑏) 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑃2𝑛 (𝜔𝑡) = [ 𝑠(𝜁𝑛 (𝑥 = 0)) + 𝜅2 𝑡(𝜁𝑛 (𝑥 = 0))] Υ(𝜔𝑡) (5.10𝑐) ℏ𝜔 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑄2𝑛 (𝜔𝑡) = [ 𝑟(𝜁𝑛 (𝑥 = 0)) + 𝜅2 𝑢(𝜁𝑛 (𝑥 = 0))] Υ(𝜔𝑡) (5.10𝑑) ℏ𝜔 𝑃3𝑛 (𝜔𝑡) = 𝑠(𝜁𝑛 (𝑥 = 𝑑))Υ(𝜔𝑡) (5.10𝑒) 𝑄3𝑛 (𝜔𝑡) = 𝑟(𝜁𝑛 (𝑥 = 𝑑))Υ(𝜔𝑡) (5.10𝑓) 𝑍3𝑛 (𝜔𝑡) = 𝑣(𝜂𝑛 (𝑥 = 𝑑))Λ(𝜔𝑡) (5.10𝑔) 74 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑃4𝑛 (𝜔𝑡) = [ 𝑠(𝜁𝑛 (𝑥 = 𝑑)) + 𝜅2 𝑡(𝜁𝑛 (𝑥 = 𝑑))] Υ(𝜔𝑡) (5.10ℎ) ℏ𝜔 𝑒𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑄4𝑛 (𝜔𝑡) = [ 𝑟(𝜁𝑛 (𝑥 = 𝑑)) + 𝜅2 𝑢(𝜁𝑛 (𝑥 = 𝑑))] Υ(𝜔𝑡) (5.10𝑖) ℏ𝜔 𝑒𝐹1 sin 𝜔𝑡 𝑍4𝑛 = [ 𝑣(𝜂𝑛 (𝑥 = 𝑑)) + 𝜅3 𝑤(𝜂𝑛 (𝑥 = 𝑑))] Λ(𝜔𝑡) (5.10𝑗) ℏ𝜔 2 𝑒 2 𝐹0𝑑𝑖𝑒𝑙 𝐹1𝑑𝑖𝑒𝑙 sin 𝜔𝑡 𝑒 2 (𝐹1𝑑𝑖𝑒𝑙 ) sin 2𝜔𝑡 with Υ(𝜔𝑡) = exp [−𝑖 +𝑖 ] , 𝑠(𝜁𝑛 ) = 𝐴𝑖(−𝜁𝑛 ) − 𝑖𝐵𝑖(−𝜁𝑛 ) , ℏ𝑚𝜔 3 8ℏ𝑚𝜔 3 𝑟(𝜁𝑛 ) = 𝐴𝑖(−𝜁𝑛 ) + 𝑖𝐵𝑖(−𝜁𝑛 ) , 𝑡(𝜁𝑛 ) = 𝑖𝐴𝑖′(−𝜁𝑛 ) + 𝐵𝑖′(−𝜁𝑛 ) , 𝑢(𝜁𝑛 ) = 𝑖𝐴𝑖′(−𝜁𝑛 ) − 𝐵𝑖′(−𝜁𝑛 ) , 1/3 2𝑚𝑒𝐹0𝑑𝑖𝑒𝑙 𝑒 2 𝐹0 𝐹1 sin 𝜔𝑡 𝑒 2 𝐹12 sin 2𝜔𝑡 𝜅2 = ( ) , Λ(𝜔𝑡) = exp [−𝑖 +𝑖 ],𝑣(𝜂𝑛 ) = 𝐴𝑖(−𝜂𝑛 ) − 𝑖𝐵𝑖(−𝜂𝑛 ), ℏ2 ℏ𝑚𝜔 3 8ℏ𝑚𝜔 3 2𝑚𝑒𝐹0 1/3 𝑤(𝜂𝑛 ) = 𝑖𝐴𝑖 ′ (−𝜂𝑛 ) + 𝐵𝑖′(−𝜂𝑛 ), 𝜅3 = ( ) . ℏ2 The electron transmission probability, 𝑤(𝜀, 𝑥, 𝑡) = 𝑗𝑣 (𝜀, 𝑥, 𝑡)/𝑗𝑖 (𝜀) , is the transmitted electron probability current density in the vacuum 𝑗𝑣 relative to the incident electron probability current density in the metal 𝑗𝑖 , both of which are calculated from the electron probability current density 𝑗 = 𝑖ℏ/2𝑚(𝜓∇𝜓∗ − 𝜓 ∗ ∇𝜓) . It is easy to show the time-averaged transmission probability as ∞ 1 2𝑚 𝐷(𝜀) = ∑ 𝑤𝑛 (𝜀) , 𝑤𝑛 (𝜀) = Im [𝑖 √ 2 𝐸3𝑛 |𝑇3𝑛 |2 ] , for 𝐹0 = 0, (5.11𝑎) 𝑘0 ℏ 𝑛=−∞ or ∞ |𝑇3𝑛 |2 𝑖𝜅3 𝐷(𝜀) = ∑ 𝑤𝑛 (𝜀) , 𝑤𝑛 (𝜀) = Im [ ] , for 𝐹0 ≠ 0 (5.11𝑏) 𝑘0 𝜋 𝑛=−∞ with all parameters defined above. The electron emission current density is obtained from ∞ 𝐽 = 𝑒 ∫ 𝐷(𝜀)𝑁(𝜀)𝑑𝜀 , (5.12) 0 𝑚𝑘 𝑇 𝐸 −𝜀 where 𝐷(𝜀) is given in Eq. (5.11) and 𝑁(𝜀) = 2𝜋2𝐵ℏ3 ln [1 + exp ( 𝑘𝐹 𝑇 )] is the flux of electrons 𝐵 impinging normal to the metal-dielectric interface, which is calculated from the three- dimensional (3D) free electron theory of metal [122], [123], [125], with 𝑘𝐵 being the Boltzmann’s constant and 𝑇 the temperature. 75 5.3.2 Keldysh Parameter The Keldysh parameter 𝛾 is a physical indicator of transition from multiphoton absorption to optical field tunneling in photoemission. 𝛾 is estimated as the ratio of two time scales [104], [105], 4𝜋𝑡𝑡 𝛾= , (5.13) 𝑇 where 𝑇 = 2𝜋/𝜔 is the cycle of the optical field, and 𝑡𝑡 = 𝑙/𝑣 is the time of an electron with velocity 𝑣 = √2𝑊/𝑚 tunneling through the potential barrier with a width 𝑙 , 𝑊 is the work function of the metal and 𝑚 is the electron rest mass. Following the definition of Keldysh parameter, the width of the barrier must be narrowed enough by the optical field that the escaping electrons can traverse the barrier within a fraction of half cycle of the optical field. For the case of photoemission from bare metal surfaces, with the instantaneous lower limit potential profile under a single laser field shown in Fig. 5.5(a), 𝛾 = 𝜔√2𝑚𝑊/𝑒𝐹1 . In the effective single-barrier quantum model (ESQM), we calculate the Keldysh parameter as 𝛾 = 𝜔√2𝑚𝑊𝑒𝑓𝑓 /𝑒𝐹𝑒𝑓𝑓 with 𝑊𝑒𝑓𝑓 = 𝑊 − 𝜒 and 𝐹𝑒𝑓𝑓 effective laser field determined by the slope of the effective single triangular barrier (refer to Sec. 5.2 or Ref. [47] for more details). If we follow Keldysh to calculate 𝛾 as the ratio of the tunneling time to the optical period, for photoemission from dielectric-coated metal surface, the expression of 𝛾 will depend on the laser field strength. When the applied 𝐹1 is relatively small (𝑊 − 𝑒𝑑𝐹1𝑑𝑖𝑒𝑙 > 0), the incident electron has to tunnel through the potential barrier in the vacuum, as shown in Fig. 2𝑚 5.5(b), and it is found 𝛾 = 𝜔√ 𝑊 (𝑊 + 𝑒𝑑(𝐹1 − 𝐹1𝑑𝑖𝑒𝑙 )) /𝑒𝐹1 . When the applied 𝐹1 is relatively large ( 𝑊 − 𝑒𝑑𝐹1𝑑𝑖𝑒𝑙 < 0 ), electrons inside the metal can be emitted by tunneling the only 2𝑚 potential barrier inside the dielectric, as shown in Fig. 5.5(c), and we find 𝛾 = 𝜔√ 𝑊 (𝑊 − 𝜒)/ 𝑒𝐹1𝑑𝑖𝑒𝑙 . 76 Figure 5.5 Potential profile for photoemission from (a) bare metal surface; (b) dielectric-coated metal surface under a relatively small laser field; (c) dielectric-coated metal surface under a relatively large laser field. 5.3.3 Results and Discussion Based on the theory developed in Sec. 5.3.1, we provide an analysis of the photoemission from metallic cathodes coated with dielectric. The metal is assumed to be gold, with nominal work function 𝑊0 = 5.1 eV and Fermi energy 𝐸𝐹 = 5.53 eV. The laser has a wavelength of 800 nm, corresponding to the photon energy of 1.55 eV. These are the default properties of the metal and laser respectively, unless prescribed otherwise. A. Effects of dielectric properties on photoemission from a flat metal surface Figure 5.6 shows the effects of coating dielectric properties (i.e., thickness d, relative permittivity 𝜀𝑑𝑖𝑒𝑙 , and electron affinity 𝜒) on the electron transmission probability from a flat metal surface. Since most of the photoemission occurs with electron initial energies near Fermi level at ambient temperature, the free electrons inside the metal are assumed to have an initial energy 𝜀 = 𝐸𝐹 . The electron transmission probability through the nth channel 𝑤𝑛 (𝜀 = 𝐸𝐹 ) calculated from Eq. (5.11), for dielectrics of different thickness 𝑑, relative permittivity 𝜀𝑑𝑖𝑒𝑙 , and electron affinity 𝜒, is plotted in Figs. 5.6(a), 5.6(b), and 5.6(c), respectively. The laser has a field strength of 𝐹1 = 5 V/nm. The dc electric field is 𝐹0 = 0. It is found that the dominant photoemission is through four-photon absorption (n = 4) under the fields provided, regardless of the dielectric properties. By checking the Keldysh parameter 𝛾, it is found that 𝛾 ≥ 3.59 for all the cases in Figs. 5.6(a), 5.6(b), and 5.6(c), indicating the multiphoton absorption process. The dominant channel, n = 4, is consistent with the ratio of the work function to the photon energy 〈𝑊/ℏ𝜔〉, where 〈 〉 represents the next nearest integer to the value inside the bracket. 𝑤𝑛 (𝜀 = 𝐸𝐹 ) increases with decreasing 𝜀𝑑𝑖𝑒𝑙 or increasing 𝜒, due to the narrowed or reduced barrier in the 77 dielectric, which is similar to dc field emission from thin dielectric coated surfaces [79]. However, 𝑤𝑛 (𝜀 = 𝐸𝐹 ) has no clear monotonic dependence on 𝑑, as shown in Fig. 5.6(a). The electron transmission probability 𝐷(𝜀 = 𝐸𝐹 ), which is a sum of 𝑤𝑛 over all channels, is presented as a function of dielectric thickness, relative permittivity, and electron affinity in Figs. 5.6(d), 5.6(e), and 5.6(f), respectively, under various laser field strengths 𝐹1 . It is obvious that the transmission probability increases when the laser field strength increases. In Fig. 5.6(d), the transmission probability shows approximately periodic peaks with respect to the dielectric thickness. These peaks are due to resonance in the quantum interference [201] between electron waves transmitted to and reflected from the dielectric-vacuum interface, which forms constructive interference when the dielectric is of a particular thickness (see Figs. 5.7 and 5.9). As the laser field strength increases, peaks on the curves shift towards larger thicknesses, as indicated by the gray dotted line in Fig. 5.6(d). The physics behind this shift lies in that the wavelength of the electron waves inside the dielectric increases with the laser field (see Fig. 8), which can also be indicated by the wavenumber √2𝑚𝐸2𝑛 /ℏ2 , with 𝐸2𝑛 = 𝜀 + 𝑛ℏ𝜔 − 𝑈𝑝2 − 2 (𝐸𝐹 + 𝑊 − 𝜒) and 𝑈𝑝2 = 𝑒 2 (𝐹1𝑑𝑖𝑒𝑙 ) /4𝑚𝜔2 . Figure 5.6(e) shows that the transmission probability 𝐷(𝜀 = 𝐸𝐹 ) decreases with the relative permittivity of the dielectric, 𝜀𝑑𝑖𝑒𝑙 , for a given laser field, due to the smaller field 𝐹1𝑑𝑖𝑒𝑙 = 𝐹1 /𝜀𝑑𝑖𝑒𝑙 inside the dielectric. When 𝜀𝑑𝑖𝑒𝑙 is large, e.g., 𝜀𝑑𝑖𝑒𝑙 > 2, 𝐹1𝑑𝑖𝑒𝑙 inside the dielectric would be relatively smaller for a given 𝐹1 , thus the incident electron would see a double-triangular potential barrier before emission (see Fig. 5.4), yielding a rapidly decreasing slope in 𝐷(𝜀 = 𝐸𝐹 ) vs 𝜀𝑑𝑖𝑒𝑙 . When 𝜀𝑑𝑖𝑒𝑙 is small (<2), 𝐹1𝑑𝑖𝑒𝑙 would be larger, such that the incident electron would see only a single-triangular barrier inside the dielectric (since the barrier at the dielectric-vacuum interface would be below the electron initial energy level), yielding a smaller slope. The smaller slope is particularly obvious for 𝐹1 = 6, 8, and 10 V/nm when 𝜀𝑑𝑖𝑒𝑙 < 2. The trends of these curves are found to follow closely those of “area under the curve” in the potential barrier (i.e., WKBJ approximation, see Fig. 5.10). Figure 5.6(f) shows the effect of the dielectric electron affinity on photoelectron transmission probability, with 𝜀𝑑𝑖𝑒𝑙 = 2, and 𝑑 = 1 nm. 𝐷(𝜀 = 𝐸𝐹 ) increases with 𝜒, due to the lowering of the potential barrier. There appear distinct resonance peaks on each curve for a given 𝐹1 due to quantum interference. The peaks shift towards a larger 𝜒 as the laser field strength 78 increases, which is indicated by the gray dotted lines in Fig. 5.6(f). Note that the model recovers photoemission from bare metal surfaces [38], [125] when 𝑑 = 0 or when 𝜒 = 0 and 𝜀𝑑𝑖𝑒𝑙 = 1 (i.e., vacuum). Figure 5.6 Effects of dielectric properties on photoelectron transmission probability from a flat metal surface. Time-averaged electron transmission probability through the nth channel, 𝑤𝑛 , calculated from Eq. (5.11), for a dielectric of different (a) thickness 𝑑; (b) relative permittivity 𝜀𝑑𝑖𝑒𝑙 ; and (c) electron affinity 𝜒 . The photoelectron transmission probability 𝐷(𝜀 = 𝐸𝐹 ) calculated from Eq. (5.11), as a function of dielectric properties (d) thickness 𝑑; (e) relative permittivity 𝜀𝑑𝑖𝑒𝑙 ; and (f) electron affinity 𝜒, for various laser field strengths. The dc electric field F0 = 0. The resonance peaks shown in Fig. 5.6(d) are due to the electron wave interference inside the dielectric [79]. In Fig. 5.7, the instantaneous electron waves at 𝑡 = 0, including the wave 𝑡 traveling to the dielectric-vacuum interface 𝜓𝐼𝐼 (blue line), the wave reflected from the 𝑟 dielectric-vacuum interface 𝜓𝐼𝐼 (orange line), as well as the combined wave 𝜓𝐼𝐼 (gray line), inside the dielectric of various thicknesses, are plotted. A constructive interference is shown in Fig. 5.7(a) with 𝑑 = 2.45 nm where the resonance peak appears in Fig. 5.6(d) for 𝐹1 = 6 V/nm. The incident wave and reflected wave are almost in phase, especially at a distance close to the 79 dielectric-vacuum interface. Figure 5.7(b) plots the electron waves for 𝑑 = 3.8 nm, which forms a valley point in Fig. 5.6(d) for 𝐹1 = 6 V/nm. The incident wave and reflected wave are ~100 degree out of phase, which results in a small transmission probability. Figure 5.7 Electron waves inside the dielectric of thicknesses: (a) 𝑑 = 2.45 nm (peak on the curve for 𝐹1 = 6 V/nm in Fig. 5.6(d)); and (b) 3.8 nm (valley on the curve for 𝐹1 = 6 V/nm in 𝑡 Fig. 5.6(d)). The blue curve represents the wave traveling to the dielectric-vacuum interface 𝜓𝐼𝐼 . 𝑟 The orange curve represents the wave reflected from the dielectric-vacuum interface 𝜓𝐼𝐼 . The 𝑡 𝑟 gray curve represents the combination of incident and reflected waves 𝜓𝐼𝐼 = 𝜓𝐼𝐼 + 𝜓𝐼𝐼 . The laser field strength is 6 V/nm. The dielectric has 𝜒 = 1 eV and 𝜀𝑑𝑖𝑒𝑙 = 2. Note that the wavelength from peak to peak may change slightly for a dielectric of certain thicknesses under a given laser field, as shown in Fig. 5.8(a). The wavelength varies with dielectric properties, such as thickness shown in Fig. 5.8(a) and 5.8(b). The laser field strength also has effects on the wavelength. As 𝐹1 increases, the wavelength increases, as demonstrated in Figs. 5.8(a), 5.8(c) and 5.8(d). 80 Figure 5.8 Electron waves for (a) 𝐹1 = 6 V/nm and 𝑑 = 6 nm; (b) 𝐹1 = 6 V/nm and 𝑑 = 4 nm; (c) 𝐹1 = 2 V/nm and 𝑑 = 6 nm ; and (d) 𝐹1 = 10 V/nm and 𝑑 = 6 nm . The blue curve 𝑡 represents the wave traveling to the dielectric-vacuum interface 𝜓𝐼𝐼 . The orange curve represents 𝑟 the wave reflected from the dielectric-vacuum interface 𝜓𝐼𝐼 . The gray curve represents the 𝑡 𝑟 combination of incident and reflected waves 𝜓𝐼𝐼 = 𝜓𝐼𝐼 + 𝜓𝐼𝐼 . The double arrow line indicates the wavelength between peaks. The dielectric has 𝜒 = 1 eV and 𝜀𝑑𝑖𝑒𝑙 = 2. The incident wave and reflected wave inside the dielectric form constructive interference by approximately satisfying the condition, 𝑘𝑑 = 𝑛𝜋 or 2𝑑 = 𝑛𝜆𝑒 , with 𝑛 = 1, 2, 3 … (5.14) where 𝑘 and 𝜆𝑒 are the electron wavenumber and the wavelength in the dielectric respectively, and 𝑑 is the dielectric thickness. Using this condition, with an approximately constant “wavelength” 𝜆𝑒 = 1.7 nm obtained from Fig. 5.8(a), we can predict the dielectric thickness at which interference occurs in the dielectric. The comparison of the predicted resonance thicknesses by Eq. (5.14) and the ones from the quantum model which are denoted as blue dots in Fig. 5.9(a), is provided in Fig. 5.9(b). Those two results show very good quantitative agreement. 81 Figure 5.9 (a) Electron transmission probability as a function of dielectric thickness under various laser field strengths, same as Fig. 5.6(d). Blued dots indicate the resonance peaks for 𝐹1 = 6 V/nm. (b) A comparison of the predicted resonance dielectric thickness by Eq. (5.14) and resonance dielectric thickness obtained from Fig. 5.6(d) or Fig. 5.9(a) here. The area enclosed by the potential profile and the electron initial energy provides a good explanation for the slope variation in Fig. 5.6(e) for electron transmission probability as a function of dielectric relative permittivity. The area (shown as red shaded area in Fig. 5.10(a)) reads 𝑥′ 𝑆 = ∫ [𝑉(𝑥, 𝑡 = 0) − 𝜀] 𝑑𝑥 , (5.15) 0 where 𝑉(𝑥, 𝑡 = 0) is the potential barrier provided as Eq. (5.3) at 𝑡 = 0 with 𝐹0 = 0, 𝜀 is the electron initial energy, and 𝑥′ is such that 𝑉(𝑥, 𝑡 = 0) − 𝜀 = 0. Figures 5.10(b) and 5.10(c) plot the negative area S and exponential function of normalized negative S as a function of dielectric relative permittivity under various laser fields. Both plots show a similar trend with Fig. 5.6(e), especially the slope change for 𝐹1 = 8 and 10 V/nm. It is because electrons would face the only triangular barrier inside the dielectric when 𝜀𝑑𝑖𝑒𝑙 is small and 𝐹1 is large enough, since the barrier at the dielectric-vacuum interface would be below 𝜀. 82 Figure 5.10 (a) Energy diagram for photoemission from dielectric-coated metal surfaces. The red shaded area denotes the area enclosed by the potential profile (at the time instant of peak laser field) and the electron initial energy 𝜀 . (b) Negative S as a function of dielectric relative permittivity under various laser field strengths. (c) Exponential function of negative normalized S as a function of dielectric relative permittivity under various laser field strengths. The area S is normalized to S at 𝜀𝑑𝑖𝑒𝑙 = 8 for 𝐹1 = 2 V/nm. The electron transmission probability 𝐷(𝜀 = 𝐸𝐹 ) is shown as a function of the laser field strength 𝐹1 under different dielectric thickness, relative permittivity, and electron affinity in Figs. 5.11(a), 5.11(b), and 5.11(c), respectively. The applied dc electric field 𝐹0 is 0. The electron transmission probability is well scaled as 𝐷(𝜀 = 𝐸𝐹 ) ∝ 𝐹12𝑛 , with 𝑛 = 4, in the small laser field range. The sharp slope change of the curves is observed at 𝐹1 ≈ 11.9 V/nm (see the vertical blue dashed lines in Fig. 5.11) due to the channel closing effect [38], which indicates the transition from multiphoton absorption to optical field emission. For the chosen parameter ranges, it is found that the dielectric coating has little effects on the laser field strength at which the transition from multiphoton absorption to optical field emission occurs. 83 Figure 5.11 The photoelectron transmission probability 𝐷(𝜀 = 𝐸𝐹 ) calculated from Eq. (5.11), as a function of laser field strength 𝐹1 for different dielectric (a) thickness 𝑑 ; (b) relative permittivity 𝜀𝑑𝑖𝑒𝑙 ; and (c) electron affinity 𝜒. The dc electric field F0 = 0. Figure 5.12 shows the effects of dielectric properties on photoemission current density. The photoemission current density 𝐽, calculated from Eq. (5.12), is plotted as a function of dielectric thickness 𝑑 in Fig. 5.12(a), for laser field strengths from 2 to 10 V/nm. 𝐽 decreases with 𝑑 for 0 < 𝑑 < 1 nm. The current density is almost constant for 𝑑 > 1 nm, while there appear some slight resonance peaks, such as the orange dots on the curve for 𝐹1 = 2 V/nm. These features are similar to those of dc field emission from dielectric coated surfaces (cf. Fig. 5 of Ref. [79]). The resonances are not as strong as those in Fig. 5.6(d) for 𝐷(𝜀 = 𝐸𝐹 ) vs 𝑑 . This is because photoemission current includes the electron emission from all incident energy levels in the metal, which, in combination, smoothens the curve and reduces the strong emission peaks from a single initial energy level. Note that in our model, electron-phonon scattering and electron-electron scattering effects in dielectric are not considered. According to the experiment in [202], a mean free path of a few nanometers has been observed for photoexcited electrons in dielectrics [202], [203]. Therefore, for dielectric thickness smaller than the mean free path, the scattering effect inside the coating may not be important. Figure 5.12(b) shows 𝐽 as a function of relative permittivity 𝜀𝑑𝑖𝑒𝑙 . The curves exhibit two distinct slopes in the semilog scale plot, which can be inferred from the transmission probability in Fig. 5.6(e). Figure 5.12(c) presents the effects of dielectric electron affinity on photoemission current density. Generally, emission current density increases with electron affinity for a given field strength, due to the lowering of the surface potential barrier. Mild resonant peaks in 𝐽 are also observed as 𝜒 changes. 84 Figure 5.12 The photoelectron emission current density 𝐽 from a dielectric-coated flat metal surface as a function of dielectric properties (a) thickness 𝑑; (b) relative permittivity 𝜀𝑑𝑖𝑒𝑙 ; and (c) electron affinity 𝜒 , for laser field strengths from 2 to 10 V/nm. The dc field 𝐹0 = 0 , and temperature 𝑇 = 300 K. Figure 5.13 The photoemission current density 𝐽 from a dielectric-coated flat metal surface as a function of dielectric thickness 𝑑 under a different (a) relative permittivity 𝜀𝑑𝑖𝑒𝑙 ; (b) electron affinity 𝜒. The dc electric field F0 = 0. It is interesting to find that a flat cathode surface with dielectric coating can emit a larger current density than an uncoated case, when the dielectric has sufficiently small relative permittivity 𝜀𝑑𝑖𝑒𝑙 or large electron affinity 𝜒, which would result in a narrowed or lowered potential barrier. Figures 5.13(a) and 5.13(b) provide examples of such cases, as shown in the region above the gray dashed lines. Similar electron emission enhancement phenomenon is also demonstrated for field emission from dielectric-coated surfaces [79]. It should be pointed out that scattering of electrons with phonons, impurities, and even with other electrons inside the dielectric is not considered in the model. Photoelectron emission from ultrathin oxide covered 85 devices shows an exponential attenuation behavior for the relatively thicker oxide layer (2.5~15.3 nm), with the dominant mean-free-path of the photoexcited electrons inside the SiO2 ~1.2 nm [202]. Therefore, for dielectric thicker than the mean-free-path, electron scattering effects cannot be ignored. B. Effects of dc field on photoemission Figure 5.14 shows the effects of dc electric field on photoemission from dielectric-coated metal surfaces. The transmission probability through different n channels from initial energy 𝜀 = 𝐸𝐹 under different 𝐹0 is shown in Fig. 5.14(a). As 𝐹0 increases, the dominant emission channel shifts to smaller n. The narrowed and lowered (due to Schottky effect) surface potential barrier by the static field enables more photoemission mechanisms, such as photo-assisted field emission (1 < 𝑛 < 4), multiphoton emission (𝑛 < 0), and direct tunneling (𝑛 = 0). The electron transmission probability is greatly enhanced by the dc field. The electron transmission probability 𝐷(𝜀 = 𝐸𝐹 ) is plotted as a function of dc field 𝐹0 in Fig. 5.14(b) for different laser field strengths. The combined dc field and laser field result in an emission probability significantly larger than that from either field alone. When 𝐹1 = 0, the electron transmission probability recovers the dc field emission from dielectric coated surfaces [79]. Figure 5.14(c) shows the emission current density as a function of dc field under various laser field strengths. The shape of the emission current density is similar to the transmission probability from 𝜀 = 𝐸𝐹 in Fig. 5.14(b). For 𝐹0 ≲ 12 V/nm, the slope of the curves varies with the laser field strength 𝐹1 , which indicates that n-photon-assisted field tunneling dominates in this range, where 𝑛 becomes smaller as 𝐹0 increases. When 𝐹0 ≳ 12 V/nm, the slopes of the curves for all four cases are almost the same, because the dominant emission becomes dc field tunneling. 86 Figure 5.14 Effects of dc field on photoemission from dielectric-coated metal surfaces. (a) Photoemission probability through different n-channels from initial energy level 𝜀 = 𝐸𝐹 under various dc fields with the laser field strength 𝐹1 = 5 V/nm. (b) Electron transmission probability 𝐷(𝜀 = 𝐸𝐹 ), and (c) Electron emission current density 𝐽, as a function of dc field 𝐹0 under various laser field strengths. The dashed line in (b) is for calculation using Eq. (8) of Ref.[79]. The coating has 𝜀𝑑𝑖𝑒𝑙 = 2, 𝜒 = 1 eV, and 𝑑 = 1 nm. 5.4 Comparison of Models In this section, we compare the results of our exact analytical model with the effective single-triangular-barrier quantum model (ESQM) [47] and modified Fowler-Nordheim (FN) equation [183], [184] for photoemission from dielectric-coated metal surfaces. The description of ESQM is provided in Sec. 5.2.2. A short account of the modified FN equation is provided herein. 5.4.1 Modified Fowler-Nordheim Equation for Double-triangular Barrier Following the procedure in [183], a modified Fowler-Nordheim equation is formulated. For the scenario of an ultrathin dielectric on a metal surface, a double step barrier is formed. If the zero of energy is taken at Fermi level, the lowermost barrier profile bent by the laser field reads, 0, 𝑥 < 0, 𝑉(𝑥, 𝑡) = { 𝑊𝑒𝑓𝑓 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡), 0 ≤ 𝑥 < 𝑑, (5.16) 𝑊 + 𝑒𝑑 (𝐹1 (𝑡) − 𝐹1𝑑𝑖𝑒𝑙 (𝑡)) − 𝑒𝐹1 (𝑡)𝑥, 𝑥 ≥ 𝑑, where 𝑊 is the nominal work function of the metal; 𝜒 and 𝑑 are the electron affinity and the thickness of the dielectric; 𝑊𝑒𝑓𝑓 = 𝑊 − 𝜒 is the effective work function at metal-dielectric interface; 𝐹1 and 𝐹1𝑑𝑖𝑒𝑙 are the laser field strengths in the vacuum and in the dielectric respectively; and 𝑒 is the positive elementary charge. Note that dc field is not considered in this modified Fowler-Nordheim equation. 87 In the current emission calculation, only the positive half cycle of the laser field is considered. According to the WKBJ approximation, we have the transmission probability, 𝐷(𝜀, 𝐹1 ) = exp[𝑄(𝜀, 𝐹1 )] , (5.17) 𝑥′ where 𝑄(𝜀, 𝐹1 ) = −2 ∫0 √2𝑚[𝑉(𝑥) − 𝜀]/ℏ2 𝑑𝑥 , 𝑚 is the effective electron mass, ℏ is the reduced Planck’s constant, and 𝑥′ is such that 𝑉(𝑥) − 𝜀 = 0. For simplicity, 𝑚 is set to equal the electron rest mass in all three regions. It is easy to show that, 4 √2𝑚 3 3 𝑑𝑖𝑒𝑙 𝑑𝑖𝑒𝑙 𝑄(𝜀, 𝐹1 ) = − [(𝑊𝑒𝑓𝑓 − 𝜀) −𝐻(𝑊𝑒𝑓𝑓 − 𝑒𝐹1 𝑑 − 𝜀)(𝑊𝑒𝑓𝑓 − 𝑒𝐹1 𝑑 − 𝜀)2 ] 2 3 𝑒ℏ𝐹1𝑑𝑖𝑒𝑙 4 √2𝑚 3 − 𝐻(𝑊 − 𝑒𝐹1𝑑𝑖𝑒𝑙 𝑑 − 𝜀)(𝑊 − 𝑒𝐹1𝑑𝑖𝑒𝑙 𝑑 − 𝜀)2 . 3 𝑒𝐹1 ℏ (5.18) Since most of the emission comes from the immediate neighborhood of the Fermi level, we replace 𝑄(𝜀, 𝐹1 ) with the first two terms of the Taylor expansion for 𝑄(𝜀, 𝐹1 ) at 𝜀 = 𝐸𝐹 = 0, which reads, 3 4 √2𝑚𝑊 2 √2𝑚𝑊 𝑄(𝜀, 𝐹1 ) = − 𝑑𝑖𝑒𝑙 𝐶+2 𝐵𝜀 (5.19) 3 𝑒ℏ𝐹1 (𝑡) 𝑒ℏ𝐹1𝑑𝑖𝑒𝑙 (𝑡) with 𝑊𝑒𝑓𝑓 𝑊𝑒𝑓𝑓 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑 𝐵=√ − 𝐻(𝑊𝑒𝑓𝑓 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑)√ 𝑊 𝑊 1 𝑊 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑 + 𝐻(𝑊 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑) √ 𝜀𝑑𝑖𝑒𝑙 𝑊 and 3 𝑊𝑒𝑓𝑓 3/2 𝑊𝑒𝑓𝑓 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑 2 𝐶=( ) − 𝐻(𝑊𝑒𝑓𝑓 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑) [ ] 𝑊 𝑊 3 1 𝑊 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑 2 + 𝐻(𝑊 − 𝑒𝐹1𝑑𝑖𝑒𝑙 (𝑡)𝑑) [ ] , 𝜀𝑑𝑖𝑒𝑙 𝑊 where 𝜀𝑑𝑖𝑒𝑙 is the relative permittivity of the dielectric, and 𝐻(𝑥) is the Heaviside step function. 𝐹1𝑑𝑖𝑒𝑙 = 𝐹1 /𝜀𝑑𝑖𝑒𝑙 is used for a perfectly flat dielectric-vacuum interface. The H functions arise from different tunneling scenarios under different field strengths (cf. Fig. 5 in [184]). 88 The emission current density can be calculated from ∞ 𝐽 = 𝑒 ∫ 𝑁(𝜀, 𝑇)𝐷(𝜀, 𝐹1 )𝑑𝜀 . (5.20) −∞ 𝑚𝑘 𝑇 𝐸 −𝜀 𝑚 At a low temperature, 𝑁(𝜀, 𝑇) = 2𝜋2𝐵ℏ3 ln [1 + exp ( 𝑘𝐹 𝑇 )] ≈ 2𝜋2ℏ3 (𝐸𝐹 − 𝜀). We finally yield 𝐵 the modified Fowler-Nordheim equation for the double-triangular barrier, 2 𝑒 3 𝐹1𝑑𝑖𝑒𝑙 (𝑡) 4√2𝑚 3 𝐽(𝑡) = 𝐻[𝐹1 (𝑡)] exp [− 𝑊 2 𝐶] (5.21) 16𝜋 2 ℏ𝑊𝐵 2 3𝑒ℏ𝐹1𝑑𝑖𝑒𝑙 (𝑡) The time-averaged photoemission current density is 2𝜋 𝜔 𝜔 𝐽= ∫ 𝐽(𝑡)𝑑𝑡 , (5.22) 2𝜋 0 where 𝜔 is the angular frequency of the laser. 5.4.2 Comparison Results for 1D Flat Cathodes Figure 5.15 shows the comparison of the three models for photoemission from 1D flat cathodes. The electron emission probability 𝐷(𝜀 = 𝐸𝐹 ) as a function of the laser field strength 𝐹1 is plotted in Fig. 5.15(a). It can be seen that, for flat surfaces with coatings, the ESQM overestimates the photoemission probability, but the modified FN equation underestimates the photoemission probability, as compared to the exact double-triangular barrier model. ESQM models the double barrier using an effective single triangular barrier based on WKBJ approximation [47], where the effective barrier height is chosen at 𝑊𝑒𝑓𝑓 = 𝑊0 − 𝜒. When 𝐹1 < 6 V/nm, the dominant emission process is multi-photon absorption, since the curve scales as 𝐷 ∝ 𝐹12𝑛 from both ESQM and the exact double-barrier quantum model. Here, 𝑛 = 3 for ESQM and 𝑛 = 4 for the exact double-barrier model, which is determined by the ratio of the barrier height 𝑊𝑒𝑓𝑓 to the photon energy (ℏ𝜔 = 1.55 eV for 800 nm laser) in each model, with 𝑊𝑒𝑓𝑓 = 𝑊0 − 𝜒 = 4.1 eV for ESQM and 𝑊𝑒𝑓𝑓 = 𝑊0 = 5.1 eV for the exact double-barrier model. The smaller effective potential barrier used in ESQM also explains the greater transmission probability by ESQM. The regime of multiphoton absorption is further confirmed by the Keldysh parameter, 𝛾 = 4.75 (Exact) and 𝛾 = 5.68 (ESQM) for 𝐹1 = 6 V/nm. The abrupt slope change at 𝐹1 ≈ 12 V/nm from the exact model is due to the channel closing effect [38], [119] (see Fig. 5.11). The underestimation of 𝐷(𝜀 = 𝐸𝐹 ) from FN equation is expected, because FN equation takes account of optical field tunneling only, which is significantly smaller than other n-photon 89 processes in the multiphoton absorption regime. For 13 < 𝐹1 < 15 V/nm, the curve for FN equation overlaps with that for the exact model, as the emission enters the optical field tunneling regime. Figure 5.15(b) shows emission current density as a function of the laser field strength 𝐹1 , showing similar trends as the emission probability in Fig. 5.15(a). The curves from the three models converge as the laser field increases to 15 V/nm, when the strong optical field tunneling becomes dominant. Figure 5.15 Comparison of the exact quantum model with the effective single-barrier quantum model (ESQM) and modified Fowler-Nordheim (FN) equation for photoemission from 1D flat dielectric-coated metal surfaces. (a) Transmission probability 𝐷(𝜀 = 𝐸𝐹 ) , and (b) emission current density 𝐽, as a function of the laser field strength 𝐹1 in the vacuum. Here, we use 𝜀𝑑𝑖𝑒𝑙 = 2, 𝜒 = 1 eV, and d = 1nm. The dc electric field F0 = 0. 5.4.3 Comparison Results for Pyramidal Cathodes It has been demonstrated that a pyramid-shaped plasmonic resonant photoemitter coated with an atomically thick dielectric can produce an emission current of orders of magnitude larger than a bare photoemitter [47]. Although ESQM was used to estimate the photoemission current and has been verified by comparing it with modified FN equation in Ref.[47], it is desirable to calculate photoemission from the exact analytical model. Figure 5.16 presents emission current density 𝐽, calculated from the exact double-barrier quantum model, ESQM, and modified FN equation for photoemission from the pyramid-shaped gold emitter with SiO2 coating, as a function of the externally applied laser field strength 𝐹𝑒𝑥𝑡 . Full wave optical simulation shows an approximately linearly decaying laser field inside the dielectric at the resonance wavelength of 608 nm, with the maximum field enhancement factor 𝛽 at the metal-dielectric interface (𝑥 = 0 in 90 Fig. 5.4). To accommodate to the double-triangular barrier quantum model and the modified FN equation, the laser field inside the coating is assumed uniform, with the field strength being the one at the metal-dielectric interface, and the laser field in the vacuum is assumed to be that at the dielectric-vacuum interface, i.e., 𝐹1𝑑𝑖𝑒𝑙 = 𝛽(𝑥 = 0)𝐹𝑒𝑥𝑡 , and 𝐹1 = 𝛽(𝑥 = 𝑑)𝐹𝑒𝑥𝑡 . Those three models manifest quantitatively good agreement for 𝐹𝑒𝑥𝑡 > 0.05 V/nm , where the emission enters strong field tunneling regime [47]. This transition is also indicated from the value of Keldysh parameter approaching unity, 𝛾 = 1.89 (Exact) and 𝛾 = 2.46 (ESQM) for 𝐹𝑒𝑥𝑡 = 0.05 V/nm . For 0.01 < 𝐹𝑒𝑥𝑡 < 0.05 V/nm , emission current densities calculated from both ESQM and FN equation are smaller than that from the exact double-barrier model. Figure 5.16 Photoemission current density 𝐽, calculated from the exact double barrier quantum model (Exact), effective single-barrier quantum model (ESQM), and modified Fowler-Nordheim equation (FN equation), as a function of the externally applied laser field strength 𝐹𝑒𝑥𝑡 for a pyramid-shaped plasmonic resonant photoemitter with SiO2 coating [47]. Here, the coating thickness d = 1nm, coating electron affinity 𝜒 = 0.9 eV, coating relative permittivity 𝜀𝑑𝑖𝑒𝑙 = 2.25, field resonance wavelength is at 608 nm, field enhancement factor at metal-dielectric interface 𝛽(𝑥 = 0) = 200, and field enhancement factor at dielectric-vacuum interface 𝛽(𝑥 = 𝑑) = 44 [47]. The dc electric field F0 = 0. One may expect that the assumption of the uniform laser field inside the coating being the maximum field at metal-dielectric interface would result in an overestimate of the emission current in the small field regime. However, calculation from the exact double-barrier model with 𝐹1𝑑𝑖𝑒𝑙 determined by the slope of the line connecting the peaks of the potential barrier at two interfaces (see Fig. 5.17), shows that emission current densities from ESQM and FN equation are still smaller for 𝐹𝑒𝑥𝑡 < 0.03 V/nm, despite the fact that such an approximation of 𝐹1𝑑𝑖𝑒𝑙 would overestimate the potential barrier, thus underestimating photoemission from the exact model. Therefore, applying ESQM to a dielectric-coated plasmonic resonant Au photoemitter [47] may 91 underestimate the photoemission current for 𝐹𝑒𝑥𝑡 < 0.03 V/nm, and the exact double-triangular barrier model may be used to give a more accurate estimation of photoemission from coated photoemitters. Figure 5.17 (a) Potential barrier profiles. Black dotted line: original potential profile with laser field profile obtained from full wave optical simulation [47]; red dotted line: approximated potential profile with the uniform laser field in the dielectric determined by the slope of the line; blue dotted line: approximated potential profile with the uniform laser field in the dielectric of the field at the metal-dielectric interface. The blue dotted line is used in the FN equation (Fig. 5.16 and 5.17(b)) and exact quantum model (Fig. 5.16). (b) Photoemission current density 𝐽, calculated from the exact double-barrier quantum model, effective single-barrier quantum model (ESQM), and modified Fowler-Nordheim equation (FN equation), as a function of the externally applied laser field strength 𝐹𝑒𝑥𝑡 for a pyramid-shaped photoemitter with SiO2 coating. The calculation in the exact model uses the approximate potential profile shown as the red dotted line in (a). The modified FN equation uses the approximate potential profile shown as the blue dotted line in (a). 5.5 Concluding Remarks In this chapter, we use our effective single-barrier quantum model (ESQM) to 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 great enhancement is due to the combined effects of the significantly localized enhanced plasmon resonant fields and the reduced potential barrier induced by the coating. We also construct an analytical model for photoemission from metal surfaces coated with a nanoscale dielectric, by exactly solving the one-dimensional (1D) time-dependent Schrödinger 92 equation subject to a double-triangular barrier. The model manifests various electron emission mechanisms, i.e., multiphoton photoemission, static field tunneling, photon-assisted field emission, optical field tunneling, and thermionic emission, depending on the applied fields (laser field and dc field) and temperature. The effects of dielectric properties on photoemission are investigated. It is found that a flat metal surface coated with a dielectric of smaller relative permittivity and larger electron affinity can photoemit a current density larger than the uncoated metal due to the lowered surface barrier. For dielectric coated nanoemitters, photoemission can be greatly enhanced due to the nonlinear field enhancements near the coating. Our model is compared with the effective single-barrier quantum model and the modified Fowler-Nordheim equation, for both flat cathodes and three-dimensional (3D) nanoemitters. It is found that both the effective single-barrier quantum model and the modified Fower-Nordheim equation may underestimate photoemission from the dielectric-coated nanoemitters in the multiphoton absorption regime, and that our exact model may give a more accurate estimation. In the strong-field optical tunneling regime, the three models show quantitatively good agreement. The results further confirm that plasmonic resonant tip photoemitters with nanoscale dielectric coatings may be promising for higher yield electron sources [47]. 93 CHAPTER 6 TWO-COLOR COHERENT CONTROL OF PHOTOEMISSION FROM METAL SURFACES This chapter is based on the published journal paper “Unraveling quantum pathways interference in two-color coherent control of photoemission with bias voltages,” Phys. Rev. B, vol. 106, no. 8, p. 085402, Aug. 2022, doi: 10.1103/PhysRevB.106.085402, by Yang Zhou and Peng Zhang. 6.1 Introduction Coherent control of quantum systems relies on the manipulation of quantum interference phenomena via external fields such as laser pulses. Two-color laser field consisting of a strong fundamental laser and a weak second harmonic has become an essential tool to probe and steer quantum pathways interference in the interaction with matter. Recently, two-color coherent control of photoemission from nanotips has drawn great interest, for its flexibility in manipulating electron dynamics in ultrashort temporal scale and nanometer spatial scale [84], [92], [204]–[207], which makes it applicable in spatiotemporal characterization of surface plasmon polaritons [89], [208], [209], investigation of hot-carrier dynamics [205] and strong- field photoemission [88], [210], and control of interference fringes in the momentum distribution of electron emission [90], [211]. It also opens up new opportunities in flexible control of photoemission in applications such as time-resolved electron microscopy [4], [15], [17], [19], free-electron lasers [45], [212], carrier-envelope phase detection [24], [25], [30], and emerging nanophotonic and nanoelectronic devices [26], [28], [32], [47], [207], [213]. By tuning the intensity mixture ratio and relative phase difference between the fundamental laser and its second harmonic, the photoemission current can be modulated with a contrast of up to 97.5% [84], [85], [87], [91], [92], [214]. The strong modulation of photoemission current by two-color laser is ascribed to the quantum interference between competing pathways. The excitation of two-color laser opens out multicolor quantum pathways where photons of different colors are simultaneously absorbed. Depending on the two-color intensity mixture ratio, simple two-pathway or three-pathway quantum interference model has been used to explain the scaling of the coherent signal to the second harmonic intensity in experiments [84]–[87]. However, to give satisfactory fitting to the experimental results, the simple quantum pathway model has to be modified to allow independent weights for the three channels and to account for the discrepancy in the extracted prefactors from those of independent pulses [86]. As another key tuning knob, 94 increasing dc bias field enhances electron emission but meanwhile suppresses the current modulation, as experimentally observed in [85], [86]. Nevertheless, how the dc bias field influences the weight of each pathway and interference between them is still ambiguous. Analyzing the quantum pathway model using exact quantum theory is therefore of highest interest for photoemission from metals in two-color fields and for coherent control schemes in general [86]. In this chapter, we analyze the quantum pathways interference with the exact analytical solutions of the TDSE including dc bias [91], [92]. These exact analytical quantum models for two-color laser induced photoemission show quantitative good agreement with experimental results [91], [92] and demonstrate the potential in measurement of time-resolved photoelectron energy spectra [92], [93]. The analysis explicitly shows the effects of laser fields and dc bias field on the weights of each pathway and the interference effects among them. 6.2 Analytical Quantum Models for Two-color Laser Induced Photoemission from Metal Surfaces The 1D model is illustrated in Fig. 6.1(a). Electrons with initial energy of 𝜀 emit through the metal-vacuum interface (𝑥 = 0) due to dc field 𝐹0 and two-color laser field 𝑓(𝑡) = 𝐹1 cos 𝜔𝑡 + 𝐹2 cos(2𝜔𝑡 + 𝜃), where 𝐹1 and 𝐹2 are the magnitudes of the fundamental and second harmonic electric fields, respectively, 𝜔 is the angular frequency of the fundamental laser, 𝜃 = 2𝜔𝜏 is the relative phase between the fundamental and second harmonic, with 𝜏 the corresponding time delay. For simplicity, the fields are assumed to be perpendicular to the metal surface and abruptly cut off inside the metal [25], [38], [91], [92]. The potential barrier seen by electrons inside the metal reads 0, 𝑥<0 𝜙(𝑥, 𝑡) = { , (6.1) 𝑉0 − 𝑒𝑓(𝑡)𝑥 − 𝑒𝐹0 𝑥, 𝑥 ≥ 0 where 𝑉0 = 𝐸𝐹 + 𝑊𝑒𝑓𝑓 , 𝐸𝐹 is the Fermi energy of the metal, and 𝑊𝑒𝑓𝑓 = 𝑊0 − 𝑊𝑆𝑐ℎ𝑜𝑡𝑡𝑘𝑦 is the effective work function with 𝑊0 the nominal work function and 𝑊𝑆𝑐ℎ𝑜𝑡𝑡𝑘𝑦 = 2√𝑒 3 𝐹0 /16𝜋𝜀0 the Schottky barrier lowering due to dc field 𝐹0 , 𝑒 (> 0) is the elementary charge, and 𝜀0 is the vacuum permittivity. By exactly solving the TDSE subject to the potential barrier in Eq. (6.1), the time-averaged electron transmission probability from energy level 𝜀 is obtained as (more details of the derivation, see references [91], [92]) 95 ∞ 𝐷(𝜀) = ∑ 𝑤𝑙 (𝜀) (6.2) 𝑙=−∞ where 𝑤𝑙 (𝜀) represents the electron emission through l-photon processes, with 𝑙 < 0 being multiphoton emission, 𝑙 = 0 direct tunneling, and 𝑙 > 0 multiphoton absorption processes [38], [91]–[93], [125]. The detailed expressions for 𝑤𝑙 can be found in Ref. [91] for 𝐹0 = 0 and in Ref. [92] for 𝐹0 ≠ 0. It is important to note that although 𝑙 in Eq. (6.2), as written, is referred to the number of fundamental photons ℏ𝜔, it also includes the possible processes of substituting two fundamental photons 2ℏ𝜔 with a single second-harmonic photon ℏ(2𝜔), illustrated in the three possible pathways in Fig. 6.1(a), as well as arbitrary multiples of such substitutions. Figure 6.1 (a) Energy diagram for photoemission from metal surfaces induced by two-color laser fields 𝑓(𝑡) = 𝐹1 cos(𝜔𝑡) + 𝐹2 cos(2𝜔𝑡 + 𝜃) under a dc field 𝐹0 . Red and blue arrows depict quantum pathway model, with pathway I: absorption of (4+k) fundamental photons (red arrow); pathway II: absorption of (2+k) fundamental photons and 1 second-harmonic photon (blue arrow); pathway III: absorption of 2 second-harmonic photons and k fundamental photons. (b) Electron transmission probability from initial energy 𝜀 = 𝐸𝐹 , 𝐷(𝜀 = 𝐸𝐹 ), as a function of 2𝜔 laser field 𝐹2 and phase delay 𝜃, with 𝐹1 = 2.6 V/nm and 𝐹0 = 0. Red and yellow pentagrams depict maxima and minima in 𝜃 domain for a given 𝐹2 , respectively. (c) Solid gray curve is calculated from our quantum model in Eq. (6.2), and black scatters are for visibility = 𝐷𝐼&𝐼𝐼 /(𝐷𝐼 + 𝐷𝐼𝐼 ), by fitting results with 𝐷𝑖 s obtained from Eq. (6.4). Figure 6.1(b) shows the electron emission probability from Fermi level 𝐷(𝜀 = 𝐸𝐹 ) as a function of second harmonic laser field 𝐹2 and relative phase 𝜃, with fundamental laser field 𝐹1 = 2.6 V/nm and dc field 𝐹0 = 0. The metal is assumed as gold, with 𝑊0 = 5.1 eV and 𝐸𝐹 = 5.53 eV. The fundamental laser has a wavelength of 800 nm (photon energy of 1.55 eV). For a given 𝐹2 , 𝐷(𝜀 = 𝐸𝐹 ) is a periodic function of relative phase 𝜃 with an angular frequency of 2𝜔. Maxima of 𝐷(𝜀 = 𝐸𝐹 ) for a given 𝐹2 are observed at 𝜃 ≅ 𝜋/2 (shown as red pentagrams in Fig. 6.1(b)), whereas the minima are at 𝜃 ≅ 3𝜋/2 when 𝐹2 ≤ 0.4 V/nm and at 𝜃 ≈ 𝜋 when 𝐹2 > 96 0.4 V/nm (shown as yellow pentagrams in Fig. 6.1(b)). Visibility or the modulation depth due to relative phase 𝜃, which is defined as the ratio of the difference between the maximum and the minimum of transmission probability in 𝜃 domain to the summation of them, increases as 𝐹2 increases for 𝐹2 ≤ 0.3 V/nm and keeps almost constant for 𝐹2 > 0.3 V/nm (Fig. 6.1 (c)). 6.3 Quantum Pathways Interference Model Figure 6.2 (a) Coefficients of Fourier series expansion for 𝐷(𝜏) for 𝐹2 = 0.05, 0.3, and 0.55 V/nm. (b) Fourier series coefficients at angular frequencies of 0 (n=0), 2𝜔 (n=1), and 4𝜔 (n=2) as a function of 𝐹22 . Scatters represent coefficients from Fourier series expansion of 𝐷(𝜏) from the exact quantum model, and solid curves are fitted from the quantum pathway model. (c) Decomposed electron transmission probability, with blue curves 0 frequency terms and red curves the oscillatory terms ( 2𝜔 and 4𝜔 terms). (d) Electron transmission 𝐷𝑖 through each quantum pathway (top) and their interference terms (bottom). Figure 6.2(a) shows the Fourier series coefficient 𝑐𝑛 of 𝐷(𝜀 = 𝐸𝐹 ) vs 𝜏 at multiples of second-harmonic laser frequency 2𝜔, obtained from, 𝑁 𝑐0 𝐷(𝜏) = + ∑ 𝑐𝑛 sin(𝑛(2𝜔)𝜏 + 𝜑𝑛 ) (6.3) 2 𝑛=1 97 2 𝑇 2 𝑇 with 𝑐0 = 𝑇 ∫0 𝐷(𝜏)𝑑𝜏 , 𝑐𝑛 = √𝑎𝑛2 + 𝑏𝑛2 , 𝑎𝑛 = 𝑇 ∫0 𝐷(𝜏) cos(𝑛(2𝜔𝜏)) 𝑑𝜏 , 𝑏𝑛 = 2 𝑇 2𝜋 𝑎 ∫ 𝐷(𝜏) sin(𝑛(2𝜔)𝜏) 𝑑𝜏 , 𝑇 = 2𝜔 , and 𝜑𝑛 = tan−1 ( 𝑏𝑛 ) . Three dominant components at 𝑇 0 𝑛 angular frequencies of 0, 2𝜔, and 4𝜔 are observed for 𝐹2 = 0.05, 0.3, and 0.55 V/nm. As 𝐹2 increases, 𝑐1 and 𝑐2 increase and their relative differences to 𝑐0 become smaller, indicating more contribution from high frequency components. These Fourier coefficients are shown as a function of 𝐹22 as scatters in Fig. 6.2(b). The results are fitted with the quantum interference model [84]–[87], which considers pathways I, II, and III as illustrated in Fig. 6.1(a), with the red and blue arrows representing absorption of one fundamental photon and one second-harmonic photon, respectively. The transmission probabilities of each pathway and of the interference terms between them are, 𝐷𝐼 ∝ 𝛼 𝑘 (𝐹12 )𝑘 (𝛼 4 (𝐹12 )4 ) = 𝐾𝐼 , (6.4𝑎) 𝐷𝐼𝐼 ∝ 𝛼 𝑘 (𝐹12 )𝑘 (𝜁 2 (𝐹12 )2 𝐹22 ) = 𝐾𝐼𝐼 𝐹22 , (6.4𝑏) 𝐷𝐼𝐼𝐼 ∝ 𝛼 𝑘 (𝐹12 )𝑘 (𝛽 2 (𝐹22 )2 ) = 𝐾𝐼𝐼𝐼 (𝐹22 )2 , (6.4𝑐) 𝐷𝐼&𝐼𝐼 ∝ 2√𝐷𝐼 𝐷𝐼𝐼 cos 𝜃 ∝ 𝛼 𝑘 (𝐹12 )𝑘 (2𝛼 2 𝜁(𝐹12 )3 √𝐹22 cos 𝜃) = 𝐾𝐼&𝐼𝐼 √𝐹22 cos(2𝜔𝜏) , (6.4𝑑) 𝐷𝐼&𝐼𝐼𝐼 ∝ 2√𝐷𝐼 𝐷𝐼𝐼𝐼 cos 2𝜃 ∝ 𝛼 𝑘 (𝐹12 )𝑘 (2𝛼 2 (𝐹12 )2 𝛽𝐹22 cos 2𝜃) = 𝐾𝐼&𝐼𝐼𝐼 𝐹22 cos(4𝜔𝜏), (6.4𝑒) 𝐷𝐼𝐼&𝐼𝐼𝐼 ∝ 2√𝐷𝐼𝐼 𝐷𝐼𝐼𝐼 cos 𝜃 ∝ 𝛼 𝑘 (𝐹12 )𝑘 (2𝜁𝐹12 𝛽√(𝐹22 )3 cos 𝜃) = 𝐾𝐼𝐼&𝐼𝐼𝐼 √(𝐹22 )3 cos(2𝜔𝜏), (6.4𝑓) where 𝛼, 𝜁, and 𝛽 are the weights for pathways I, II, and III, respectively, and 𝐾𝑖 is the prefactor for each term as a power function of 𝐹22 . The prefactors 𝐾𝑖 in Eq. (6.4), are shown as a function of 𝐹12 as scatters in Fig. 6.3. They are fitted by power functions of 𝐹12 , with fitting results shown as solid curves. The fitting power orders for prefactors of 𝐾𝐼 , 𝐾𝐼𝐼 , 𝐾𝐼&𝐼𝐼 , 𝐾𝐼𝐼&𝐼𝐼𝐼 , and 𝐾𝐼&𝐼𝐼𝐼 are 3.84, 2.43, 3.14, 0.9, and 1.53, respectively. These values are close to the corresponding power orders of 4, 2, 3, 1, and 2 with 𝑘 = 0 in Eq. (4), which validates the quantum pathway model and also indicates the dominant absorption of equivalent 4 fundamental photons for 𝜀 = 𝐸𝐹 . It should be noted that 𝐾𝐼𝐼𝐼 is not fitted here, since it is independent of 𝐹12 for 𝑘 = 0. Figure 6.3(a) shows that 𝐷𝐼𝐼𝐼 decreases slightly as 𝐹12 increases. With 𝑘 = 0, 𝛼, 𝜁, and 𝛽 can be calculated from 𝐾𝐼 , 𝐾𝐼𝐼 , and 𝐾𝐼𝐼𝐼 respectively for a given 𝐹1 . Those values are used to calculate 𝐾𝐼&𝐼𝐼 , 𝐾𝐼𝐼&𝐼𝐼𝐼 , and 𝐾𝐼&𝐼𝐼𝐼 , which are shown as dashed curves in 98 Fig. 6.3(c). Good agreement is achieved between the dashed curves and the scatters obtained from Eq. (6.4), which further confirms the validity of the quantum pathway model. Figure 6.3 Prefactors in Eq. (6.4), (a) 𝐾𝐼 , 𝐾𝐼𝐼 , 𝐾𝐼𝐼𝐼 , (b) and (c) 𝐾𝐼&𝐼𝐼 , 𝐾𝐼&𝐼𝐼𝐼 , and 𝐾𝐼𝐼&𝐼𝐼𝐼 as a function of 𝐹12 . Scatters: values from the fitting of Fourier series coefficients using the quantum interference model; solid curves in (a) and (b): power fitting with respect to 𝐹12 ; dashed curves in (c): values calculated with 𝛼, 𝜁, and 𝛽 from 𝐾𝐼 , 𝐾𝐼𝐼 , and 𝐾𝐼𝐼𝐼 . Equating the total transmission probability 𝐷(𝜏) in Eq. (6.3) with the sum of all the probabilities in Eqs. (6.4a)-(6.4f) enables us to extract straightforwardly the weights 𝛼, 𝜁, and 𝛽 from the Fourier coefficients. Here, by setting 𝑘 = 0 (see Fig. 6.3), the fitted 𝐾𝐼 , 𝐾𝐼𝐼 , and 𝐾𝐼𝐼𝐼 in Fig. 6.2(b) yield 𝛼 = 1.18 × 10−3 nm2 /V 2 , 𝜁 = 2.93 × 10−5 nm3 /V 3 , and 𝛽 = 3.17 × 10−4 nm2 /V 2 . With these 𝐾𝑖 weights (Fig. 6.2(b)), the transmission probability 𝐷(𝜀 = 𝐸𝐹 ) is decomposed into a dc term (0 frequency term) and an oscillatory term (consisting of 2𝜔 and 4𝜔 terms), as exemplified in Fig. 6.2(c) for 𝐹2 = 0.3 V/nm. Fourier series expansion with these fitting parameters perfectly reproduces the raw data from our quantum model obtained from Eq. (6.2). Photoemission through each of the channels in Eq. (6.4) is explicitly shown in Fig. 6.2(d). It is clear that pathways I and II in combination, form the majority of the constant baseline emission channels, around which the transmission probability oscillates with the relative phase 𝜃. The strongest interference is between pathways I and II. Interference terms, and therefore the total transmission probability, can be strongly tuned by the phase difference 𝜃, with maximum at 𝜃 ≅ 𝜋/2 and minimum at 𝜃 ≅ 1.8𝜋, as shown in Figs. 6.2(c) and 6.2(d). 99 Figure 6.4 (a) Electron transmission probability 𝐷𝑖 through each channel and (b) the corresponding normalized transmission probability 𝐷𝑖 /𝐷 at maximum of 𝐷(𝜀 = 𝐸𝐹 ) in 𝜃 domain for a given 𝐹2 . (c) 𝐷𝑖 and (d) 𝐷𝑖 /𝐷 at minimum of 𝐷(𝜀 = 𝐸𝐹 ) in 𝜃 domain for a given 𝐹2 . Here, 𝐹1 = 2.6 V/nm, and 𝐹0 = 0. The effect of second harmonic laser field 𝐹2 on the contribution of each channel to the total emission is shown in Fig. 6.4. Transmission probability 𝐷𝑖 through each channel 𝑖 at the maximum of total 𝐷(𝜀 = 𝐸𝐹 ) in 𝜃 domain for the given 𝐹2 is plotted in Fig. 6.4(a). The corresponding normalized transmission probability with respect to the total 𝐷(𝜀 = 𝐸𝐹 ) is shown in Fig. 6.4(b). When 𝐹2 ≲ 0.2 V/nm, pathway I and the interference of I&II account for more than 66% of the total emission, whereas contribution from pathway III can be neglected. The negligibility of pathway III for a low field mixture ratio (𝐹2 /𝐹1 < 7% here) is further confirmed by the visibility analysis (see Fig. 6.1(c)), which is consistent with previous work for tungsten [84]. As 𝐹2 increases, transmission probability through all channels but pathway I increases, exhibiting strong constructive interferences from I&II, II&III, and I&III. When 𝐹2 ≳ 0.35 V/nm, contributions from channels II, I&III, and II&III exceed that from pathway I. The interference II&III surpasses I&II at 𝐹2 ≅ 0.46 V/nm. Figures 6.4(c) and 6.4(d) shows the transmission probability and their normalized value at minima of 𝐷(𝜀 = 𝐸𝐹 ) in 𝜃 domain as a function of 𝐹2 . 100 The interference terms I&II, II&III, and I&III are negative, which indicate destructive interferences and greatly suppress the total transmission probability. More details of the effect of 𝐹2 is given by analyzing the dependence of 𝐷𝑖 (𝜀 = 𝐸𝐹 ) vs 𝜏 for 𝐹2 = 0.05, 0.3, and 0.55 V/nm in Fig. 6.5, which reconfirms the above observation. When 𝐹2 increases to 0.55 V/nm, emission through pathway II and interference II&III become the dominant dc and oscillatory terms respectively. Figure 6.5 Decomposed electron transmission probability by Fourier series expansion. (a), (b), and (c) 0 frequency terms for 𝐹2 = 0.05, 0.3 and 0.55 V/nm, respectively. (d), (e), and (f) oscillatory terms (2𝜔 and 4𝜔 terms) for 𝐹2 = 0.05, 0.3 and 0.55 V/nm, respectively. 𝐹1 is fixed at 2.6 V/nm. Figure 6.6 shows the effect of fundamental laser field on the modulation of quantum interference and the contribution of each pathway. As shown in Fig. 6.6(a), when 𝐹1 increases, the visibility increases and then decreases, with the maximum at 𝐹1 = 2.6 V/nm, corresponding to the intensity ratio of second harmonic to the fundamental of 1.3%, with fixed second harmonic field 𝐹2 = 0.3 V/nm and dc field 𝐹0 = 0. The transmission probability from initial energy level 𝜀 = 𝐸𝐹 as a function of relative phase 𝜃 is shown as inset in Fig. 6.6(a) under various fundamental laser fields 𝐹1 . As 𝐹1 increases, 𝐷(𝜀 = 𝐸𝐹 ) is greatly enhanced. The oscillation magnitude gradually decreases for 𝐹1 > 2.6 V/nm, consistent with the decreasing visibility in Fig. 6.6(a). 101 The decrease of the visibility is ascribed to smaller two-color laser intensity ratio, which results in relatively less contribution from pathways II and III and therefore the interference terms, as shown in Fig. 6.6(b). When 𝐹1 < 2 V/nm (or 𝐹2 /𝐹1 > 0.15), the dominant emission is through pathway III and the interference term II & III. As 𝐹1 increases, electron emission through pathway I and interference I&II increase greatly. As 𝐹1 reaches ~3.7 V/nm , the contribution from pathway I exceeds that of the interference term I&II. For 𝐹1 > 3.7 V/nm, while I&II remains the dominant interference term, its contribution to photoemission continues to decrease. Figure 6.6 (a) Visibility as a function of 𝐹1 . Gray solid curve: calculated directly from the data by the quantum model in Eq. (6.2); Black scatters: calculated from visibility = 𝐷𝐼&𝐼𝐼 /(𝐷𝐼 + 𝐷𝐼𝐼 ) with 𝐷𝐼 , 𝐷𝐼𝐼 , and 𝐷𝐼&𝐼𝐼 from Eq. (6.4) using fitted parameters; inset: transmission probability as a function of relative phase under different 𝐹1 . (b) Normalized transmission probability over the total transmission probability at maximum of 𝐷(𝜀 = 𝐸𝐹 ) in 𝜃 domain as a function of 𝐹1 . Here, we fix 𝐹2 = 0.3 V/nm and 𝐹0 = 0. 102 Figure 6.7 (a) 𝐷(𝜀 = 𝐸𝐹 ) as a function of phase delay 𝜃 under various dc fields 𝐹0 . (b) Visibility as a function of 𝐹0 under various 𝐹2 . (c) Fourier series coefficients under various 𝐹0 . (d) Normalized transmission probability over the total transmission probability at maximum of 𝐷(𝜀 = 𝐸𝐹 ) in 𝜃 domain as a function of 𝐹0 . Here, we use 𝐹1 = 2.6 V/nm and 𝐹2 = 0.3 V/nm in (a), (c), and (d). As another key knob to the coherent control of electron emission by two-color lasers, the effect of dc field is shown in Fig. 6.7. In Fig. 6.7(a), 𝐷(𝜀 = 𝐸𝐹 ) is plotted as a function of relative phase 𝜃 under various dc fields 𝐹0 . As 𝐹0 increases, transmission probability is greatly enhanced due to the lowering and narrowing of the surface potential barrier, which opens emission channels of lower order (cf. Fig. 2 in [92]). The minimum shifts from 𝜃 ≅ 1.7𝜋 to 𝜃 ≅ 𝜋, which indicates the suppression of high order 4𝜔 terms. Figure 6.7(b) shows the visibility as a function of 𝐹0 under different second harmonic laser fields with 𝐹1 = 2.6 V/nm . For 𝐹2 ≤ 0.3 V/nm, the maximum visibility is observed at 𝐹0 ≅ 0.5 V/nm, whereas for 𝐹2 ≥ 0.4 V/nm, the maximum occurs at 𝐹0 = 0. Another peak also appears at 𝐹0 ≅ 1.75 V/nm for all cases. When 𝐹0 > 1.75 V/nm, visibility decreases. As 𝐹0 reaches 3 V/nm, visibility drops to almost 0 for all cases. 103 The nonlinear dependence of visibility on dc fields can be explained by looking at the Fourier series coefficient, as shown in logarithm scale in Fig. 6.7(c). When 𝐹0 ≥ 0.5 V/nm, the high frequency component at 4𝜔 is suppressed compared to the 0 and 2𝜔 terms. As 𝐹0 further increases, the difference between the coefficient at 0 frequency and that at 2𝜔 increases. When 𝐹0 is sufficiently large, the component at 2𝜔 becomes also negligibly small compared to the 0 frequency component. Thus, the peaks around 0 - 0.5 V/nm and 1.75 V/nm in Fig. 6.7(b) can be attributed to the 4𝜔 and 2𝜔 components, respectively. Note the peaks are also observed in previous experiments (inset in Fig. 5 of [86]). Normalized transmission probability at maximum of 𝐷(𝜀 = 𝐸𝐹 ) in 𝜃 domain in Fig. 6.7(d) also confirms this observation. Since 4𝜔 term is negligible for the majority of the cases, only pathways I, II, and their interference I&II are considered here for simplicity. When 𝐹0 < 0.5 V/nm, those two pathways and the interference between them cannot account for the total emission, as the sum of those terms is smaller than 1, indicating that additional contributions from 4𝜔 term are needed in this regime. As 𝐹0 ≥ 0.5 V/nm , the sum approaches 1. The contribution from pathway I to the total emission increases, while the contribution from the other two channels decreases. It is important to note that, different from the physical meaning of absorption of 4 photons of fundamental laser in the absence of dc field, pathway I with positive 𝐹0 also includes contributions from direct tunneling, photon-assisted tunneling, and photon emission tunneling (cf. Fig. 2 in [92]), which are all captured by the prefactors with power of 𝑘 in Eq. (6.4). 6.4 Concluding Remarks In this chapter, we analyze the quantum pathways interference in two-color coherent control of photoemission using exact analytical solutions of the TDSE including dc bias. The exact theory includes the contribution from all possible quantum pathways and their interference terms. The effects of two-color laser fields and dc bias field on the weights of each pathway and interferences between them are explicitly demonstrated. It is found that increasing the intensity ratio of the second harmonic to fundamental lasers would result in stronger high frequency modulations and increased visibility more than 95%, where contributions from pathways with the replacement of two fundamental photons with one second-harmonic photon and their interferences become significant. The occurrence of maximum (or minimum) transmission with varying phase delays of the two-color lasers is due to the constructive (or destructive) interferences among the pathways. Increasing bias voltages shifts the dominant emission process 104 from multiphoton absorption to photon-assisted tunneling or direct tunneling, which sequentially decrease the weights of higher order 4𝜔 and then 2𝜔 components, resulting in two peaks in the visibility as a function of bias voltage. This study provides direct theoretical foundation to confirm the coherent emission physics of replacing two fundamental photons with one second- harmonic photon despite various possible pathways. 105 CHAPTER 7 SUMMARY AND SUGGESTED FUTURE WORKS 7.1 Summary In this thesis, we develop analytical quantum models for laser-induced photoemission from bare or dielectric-coated metal surfaces, by exactly solving the TDSE. Our models are applicable to arbitrary combination of laser properties (wavelength and field strength), metal properties (Fermi energy and work function), dc field, and dielectric properties (thickness, relative permittivity, and electron affinity). These models include various electron emission processes, i.e., multiphoton absorption photoemission, above-threshold photoemission, photo-assisted field emission, field emission, and optical field emission. The parametric dependence of those processes is discussed in detail. The quantum model for bare metal surfaces is validated by comparing with classical models, i.e., three-step model and Fowler-DuBridge model. The effects of laser wavelength (200-1200 nm) and the accompanying laser heating on the quantum efficiency of photoemission from bare metal surfaces are studied. The effects of dielectric coatings on cathode surfaces are studied. With the quantum model for two-color laser induced photoemission, we analyze the quantum pathways interference in two-color coherent control of photoemission. The effects of laser fields (the intensity mixture ratio of the second harmonic (2𝜔) to fundamental (𝜔) and their relative phase) and dc field on the weights of each pathway and the interferences among them are investigated. Using the model for photoemission from bare metal surfaces, it is found that shorter wavelength lasers can induce more photoemission from electron initial energy levels further below the Fermi level, and therefore yield a larger quantum efficiency (QE). The dc field increases QE, but it is found to have a greater impact on lasers with wavelengths close to the threshold (i.e., the corresponding photon energy is the same as the cathode work function) than on shorter wavelength lasers. By comparison with classical models, the scaling of QE of the quantum model agrees well with the other models for low intensity laser fields, even though they have different settings and assumptions. Using the model for photoemission from metal surfaces and the two temperature model, it is found that QE can be increased nonlinearly by the non-equilibrium electron heating produced by intense sub-picosecond laser pulses. This increase of QE due to laser heating is the strongest near laser wavelengths where the cathode work function is an integer multiple of the corresponding laser photon energy. The quantum model, with laser heating effects included, reproduces 106 previous experimental results, which further validates our quantum model and the importance of laser heating. For 1D flat metallic cathode surfaces with a dielectric coating, it is found the field emission current density can be larger than the uncoated case when the dielectric constant is smaller than a 𝑡ℎ certain value 𝜀𝑑𝑖𝑒𝑙 and the dielectric thickness is larger than the threshold value 𝑑𝑡ℎ [nm] ≈ 𝑡ℎ 𝜀𝑑𝑖𝑒𝑙 𝑊/𝑒𝐹 with the dielectric constant 𝜀𝑑𝑖𝑒𝑙 < 𝜀𝑑𝑖𝑒𝑙 , where 𝑊 is the work function of the cathode material, 𝐹 is the applied dc field, and 𝑒 is the elementary charge (positive). Comparison of our quantum model with a modified Fowler-Nordheim equation for double-barrier shows qualitatively good agreement. Similar to field emission, a flat metal surface with dielectric coatings can photoemit a larger current density than the uncoated case when the dielectric has a smaller relative permittivity and a larger electron affinity. Resonant peaks in the photoemission probability and emission current are observed as a function of dielectric thickness or electron affinity, due to the quantum interference of electron waves inside the dielectric. The comparison of our model with the effective single-barrier quantum model and modified Fowler-Nordheim equation, for both 1D flat cathodes and pyramid-shaped nanoemitters, shows quantitatively good agreement in the optical field tunneling regime. Our quantum model may give a more accurate evaluation of photoemission from dielectric-coated emitters in the multiphoton absorption regime. Under two-color laser field excitation, the visibility of the photoemission current can be tuned to be more than 95% by increasing the intensity ratio of the second harmonic (2𝜔) to fundamental ( 𝜔 ) lasers, which signifies the coherent emission physics of replacing two 𝜔 photons with one 2𝜔 photon. Constructive (or destructive) interferences among the pathways lead to the maximum (or minimum) emission with varying phase delay of the two-color lasers. Increasing bias voltages shifts the dominant emission process from multiphoton absorption to photon-assisted or direct tunneling, where the sequential decrease of the 4𝜔 and 2𝜔 components in current modulation leads to two peaks in the visibility as a function of dc field. 7.2 Suggested Future Works on the Improvement of the Models In this thesis, laser heating induced electron dynamics is considered by using the two- temperature model (TTM) for 450 fs laser pulses. Due to the large difference between the heat capacity of electrons and phonons, as well as picosecond-scaled electron-phonon scattering, there is a thermal nonequilibrium between electrons and the lattice [58], [135], [136]. For a laser 107 pulse of a few hundred femtoseconds to a few picoseconds, TTM is used to obtain electron temperature, with electrons in the metal following Fermi-Dirac distribution. As the laser pulse duration further decreases, TTM may fail due to the thermal nonequilibrium in both the electron and lattice systems. The microscopic kinetic theory, such as Boltzmann’s equation [62], [63], [151], needs to be used. In this thesis, the time-averaged transmission probability for continuous wave laser excitation is used for the pulsed laser photoemission. It is an excellent approximation for laser pulses of longer than 10 optical cycles [38]. Therefore, it is expected to be applicable for laser pulses of 450 fs for various laser wavelengths, with ~112.5 laser cycles for 1200 nm and ~675 laser cycles for 200 nm. As laser pulses further decrease to tens of femtoseconds or even shorter, a model for few-cycle pulsed laser induced photoemission is needed [25]. The consistent treatment of the dynamic electron energy distribution function and electron emission, induced by a laser pulse of a few to a hundred femtoseconds, will be a subject of future study. The effective electron mass in all quantum models is assumed to be electron rest mass in all regions, i.e., metal, dielectric, and vacuum. However, effective electron mass varies with materials. Metals usually have an effective electron mass close to the electron rest mass, while in semiconductors and dielectrics, electrons usually have an effective mass (with respect to electron rest mass) in the range of 0.01 to 10 [215], [216]. The external applied fields may also affect the effective electron mass. The tunneling effective mass for the electron in the SiO2 layer is found to increase slightly with applied dc bias [217]. Theoretical analysis of the effective electron mass of various semiconductors in the presence of light waves reveals that the effective electron mass increases with light intensity and wavelength [218], [219]. Including the effective electron mass in different regions into our quantum models will be another subject of future study. In the models for electron emission from dielectric-coated metal surfaces, the dielectric is assumed to be ideal, with only electron affinity and relative permittivity considered. Relative permittivity is assumed to be a constant real value all the time. For lossy material, relative permittivity can be a complex number. It is found that relative permittivity depends on the thickness of the material. Experimental and theoretical studies have shown that both real and imaginary parts of the relative permittivity of Si decrease as the thickness decreases to nanoscale (a few to ten nanometers), due to the quantum confinement effect [220], [221]. Both real and imaginary parts of the relative permittivity of Si also show their dependence on the photon energy when the thickness of Si is under a certain threshold [220], [221]. It is worthwhile to take 108 those factors into account in our quantum models. In those models for electron emission from metal surfaces with dielectric coatings, the metal, the dielectric, and the interfaces between them are all assumed to be perfectly aligned and with no defects or impurities. However, localized traps may exist in the bulk dielectric and at the metal-dielectric interface, especially when the dielectric layer becomes relatively thick [222], which may result in trapping of photoexcited electrons from metal. Trapped charges, impurities, and defects will modify the surface potential profile. Ionic defects in the dielectric in metal- dielectric-metal structure form quantum wells, resulting in enhanced tunneling current [223]. At higher fields, more electrons are emitted into the dielectric, where the space charge effect cannot be ignored [28], [32], [224]. The potential barrier is considerably modified by these factors. All these call for a numerical solution to higher-dimensional time-dependent Schrödinger equation with a realistic potential profile due to the time- and space-varying local fields, space charge, image charge, trapped charges and impurities and defects in dielectric. 7.3 Suggested Future Works on the Applications of the Models Quantum models presented in this thesis are one dimensional, with surface potential profile in (oscillatory) triangular shape. The field enhancement due to the emitter geometry, surface roughness and plasmonic effects and corresponding spatial variation are prescribed but not considered consistently in the quantum models. Triangular potential profile is still a good approximation with local field enhancement included for cathodes of various shapes [47], [79], [92]. It will be of great interest to apply quantum models with surface field enhancement included to cathodes of different geometries or nanostructures. Graphene has unique electronic properties, including high carrier mobility, ballistic transport, and linear energy dispersion relationship, which make it a promising cathode material to produce high-quality electron sources [225]. Classical models for field emission and photoemission [97], [110], [111] may not work for graphene, since they assume that the energy dispersion of the emitter has a parabolic-like function. Equations. (2.9), (4.9), and (5.12) indicate that emission current density is determined by two factors, i.e., transmission probability and supply function. Our quantum models include photoemission, field emission, and thermionic emission with laser heating induced temperature rise included. By utilizing the material-specific supply function, our model is expected to be directly applicable to graphene and other two- dimensional (2D) materials. 109 The models developed and/or utilized in this thesis can be applied to study density modulation of electron beams during their emission. Such a density modulation scheme based on optical means may open new opportunities to improve the performance of electron beam based electromagnetic sources and amplifiers [172], [226], [227]. Models in this thesis may also be extended to photoemission and transport in a nanogap, where the effects of the nearby anode will become important [123], [124], [207], [228]. The physics of photoemission in nanogaps will find applications to photodetectors [109], electrical contacts and junctions [229]–[232], and ultrafast electron microscopy [233]. High resolution scanning electron microscopy (SEM) or transmission electron microscopy (TEM) requires the electron source to be with a high brightness and a low energy spread [19], [234]. Quantum models for field/photo- emission from dielectric-coated metal surfaces indicate that the quantum well formed on the surface due to the dielectric induces quantum resonance effects. Metal surfaces with dielectric coatings of certain properties (thickness, electron affinity, and relative permittivity) can produce enhanced electron emission with narrow energy peaks on the energy spectrum (see Fig. 4.4), compared to electron emission from bare metals. An ongoing work on photoemission from metal surfaces with a nearby quantum well also observes the aforementioned quantum resonant enhanced emission. 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