Electron scattering and static field effects in high-order harmonic generation in solid systems

Electron Scattering and Static Field Effects in High-order Harmonic Generation in Solid Systems Dissertation zur Erlangung des Doktorgrades an der Fakult¨at f¨ur Mathematik, Informatik und Naturwissenschaften Fachbereich Physik der Universit¨at Hamburg vorgelegt von Chang-Ming Wang Hamburg 2021 Gutachter/innen der Dissertation: Prof. Dr. Angel Rubio Prof. Dr. Nina Rohringer Zusammensetzung der Pr¨ufungskommission: Prof. Dr. Angel Rubio Prof. Dr. Nina Rohringer Prof. Dr. Michael Potthoff Prof. Dr. Nils Huse Prof. Dr. Shunsuke Sato Vorsitzender der Pr¨ufungskommission: Prof. Dr. Michael Potthoff Datum der Disputation: 29.09.2021 Vorsitzender Fach-Promotionsausschusses PHYSIK: Prof. Dr. Wolfgang Hansen Leiter des Fachbereichs PHYSIK: Prof. Dr. G¨unter Hans Walter Sigl Dekan der Fakult¨at MIN: Prof. Dr. Heinrich Graener Abstract The strong non-linear light emission induced by a high-intensity laser in solid systems has become a research topic of immense interest over the last few years. Under such a strong laser, electrons from the material will generate photons with energy being integer multiple of that of the input laser photon. This specific light emission process is addressed as high-order harmonic generation (HHG) and has been widely studied in systems of atomic or molecular gas. With the great understanding of the mechanisms of the photon emission process, several powerful applications based on the gas-phase HHG are developed, enabling the generation of isolated short pulses [1], atomic or molecular orbital tomography [2], and real-time observation of electron dynamics [3]. Recently HHG has also been observed in solid systems using a semiconductor as a target. Considering many useful applications utilizing HHG from gas systems have been developed and widely utilized in various scientific fields, adaptation of the applications based on gas-phase HHG to the realm of solid-phase HHG has been pursued by many researchers. Whether the adaptation could be carried out highly depends on what physicists know about solid-phase HHG. As a result, a deep understanding of the dynamics and development of theoretical models are highly demanded in the community of solid-phase HHG. In this thesis we investigate solid-phase HHG under the influence of electron scattering or an additional static field in an attempt to achieve a better understanding of the underline dynamics. For the studies of electron scattering, we integrate Umklapp scattering into the generalized three-step model [4,5] and compare the results from this modified model with those from ab initio quantum simulations. This leads to our publication [6] showing that in HHG power spectra each of the multi-plateau, which originates from the band climbing [7], is dominated by the light emission from electron-hole pairs experiencing a specific number of scattering; An electron-hole pair with zero, one, and two scattering before emitting a photon mainly contributes to the first, second, and third plateaus of an HHG power spectrum, respectively. In addition, we also consider another simple modification to the generalized three-step model for treating general scattering effects in solids based on a mean-free-path approach. We find that such a simple modification could reproduce the wavelength independence of cutoff energy for solid-phase HHG, suggesting such behavior is directly related to scattering processes in solids. As for the studies on the effect of an additional static field, we add a static electric field on top of a driving laser for HHG based on a simple two-band parabolic quantum model. The i resulting HHG power spectra yield an overall lower emission intensity and static-fielddependent cutoff energy. When increasing the static field from zero, the cutoff energy will increase, reach a maximum, and then decrease to the band gap when the static field becomes as strong as the oscillating driving laser. This static-field-dependence of the cutoff energy could be described by the two competing mechanisms induced by a static field: reduced probability for overall electron-hole recombination and increased chances for recombination for some high-energy electron-hole pairs when the static field happens to align with the driving laser pushing the pairs together. In addition to the studies on dynamics of solid-phase HHG, we also present a preliminary investigation of the core electron absorption for bulk aluminum under X-ray by time-dependent density functional theory (TDDFT). The aim was to verify whether the underline theoretical model could capture the well-known absorption saturation in aluminum [8] so as to estimate the applicability of the simulation framework for solidphase HHG driven by X-ray pulses. From our simulations, the absorption saturation is indeed reproduced qualitatively. This suggests the first step of solid-phase HHG in the three-step model, namely the excitation of electrons, could be captured by ab initio simulations based on TDDFT. ii Abstraktum Die starke nichtlineare Lichtemission eines hochintensiven Lasers in Festk¨orpersystemen ist in den letzten Jahren zu einem Forschungsthema von immensem Interesse geworden. Unter einem so starken Laser erzeugen Elektronen aus dem Material Photonen mit einer Energie, die ein ganzzahliges Vielfaches der des eingegebenen Laserphotons ist. Dieser spezifische Lichtemissionsprozess wird als harmonische Erzeugung hoher Ordnung (HHG) bezeichnet und wurde in Systemen aus atomaren oder molekularen Gasen umfassend untersucht. Mit dem großen Verst¨andnis der Mechanismen des Photonenemissionsprozesses werden mehrere leistungsstarke Anwendungen basierend auf dem HHG in der Gasphase entwickelt, die die Erzeugung isolierter kurzer Pulse [1], Atomoder Molek¨ulorbitaltomographie [2] und Echtzeitbeobachtung der Elektronendynamik [3]. K¨urzlich wurde HHG auch in festen Systemen mit einem Halbleiter als Target beobachtet. In Anbetracht vieler n¨utzlicher Anwendungen, die HHG aus Gassystemen verwenden, wurden entwickelt und auf verschiedenen wissenschaftlichen Gebieten weit verbreitet verwendet, eine Anpassung der Anwendungen basierend auf GasphasenHHG an den Bereich von Festphasen-HHG wurde von vielen Forschern verfolgt. Ob die Anpassung durchgef¨uhrt werden k¨onnte, h¨angt stark davon ab, was Physiker ¨uber Festphasen-HHG wissen. Aus diesem Grund sind in der Gemeinschaft der FestphasenHHG ein tiefes Verst¨andnis der Dynamik und Entwicklung theoretischer Modelle sehr gefragt. In dieser Arbeit werden wir Festphasen-HHG unter dem Einfluss von Elektronenstreuung oder einem zus¨atzlichen statischen Feld untersuchen, um ein besseres Verst¨andnis der Unterstreichungsdynamik zu erreichen. F¨ur die Untersuchungen der Elektronenstreuung integrieren wir die Umklapp-Streuung in das verallgemeinerte dreistufige Modell [4,5] und vergleichen die Ergebnisse dieses modifizierten Modells mit denen eines ab initio Quantensimulationen. Dies f¨uhrt zu unserer Publikation [6], die zeigt, dass in HHG-Leistungsspektren jedes der Multiplateaus, die aus dem Bandanstieg stammen, [7] , wird von der Lichtemission von Elektron-Loch-Paaren dominiert, die eine bestimmte Anzahl von Streuungen erfahren; Ein Elektron-Loch-Paar mit einer Streuung von Null, Eins und Zwei vor der Emission eines Photons tr¨agt haupts¨achlich zum ersten, zweiten und dritten Plateau eines HHG-Leistungsspektrums bei. Dar¨uber hinaus betrachten wir eine weitere einfache Modifikation des verallgemeinerten dreistufigen Modells zur Behandlung allgemeiner Streueffekte in Festk¨orpern auf der Grundlage eines Mean-Free-Path-Ansatzes. Wir stellen fest, dass eine solch einfache Modifikation iii die Wellenl¨angenunabh¨angigkeit der Abschaltenergie f¨ur Festphasen-HHG reproduzieren k¨onnte, was darauf hindeutet, dass ein solches Verhalten direkt mit der Streuung in Festk¨orpern zusammenh¨angt. F¨ur die Studien zur Wirkung eines zus¨atzlichen statischen Feldes f¨ugen wir ein statisches elektrisches Feld ¨uber einem treibenden Laser f¨ur HHG hinzu, basierend auf einem einfachen parabolischen Zwei-Band-Quantenmodell. Die resultierenden HHG-Leistungsspektren ergeben eine insgesamt geringere Emissionsintensit¨at und statikfeldabh¨angige Abschaltenergie. Wenn ein statisches Feld von Null erh¨oht wird, steigt die Abschaltenergie an, erreicht ein Maximum und nimmt dann bis zur Bandl¨ucke ab, wenn das statische Feld so stark wird wie der oszillierende Antriebslaser. Diese statische Feldabh¨angigkeit der Abschaltenergie k¨onnte durch die beiden konkurrierenden Mechanismen beschrieben werden, die durch ein statisches Feld induziert werden: verringerte Wahrscheinlichkeit f¨ur die gesamte Elektron-Loch-Rekombination und erh¨ohte Chancen f¨ur die Rekombination f¨ur einige hochenergetische Elektron-Loch-Paare im statischen Feld richtet sich zuf¨allig mit dem treibenden Laser aus und dr¨uckt die Paare zusammen. Zus¨atzlich zu den Untersuchungen zur Dynamik von Festphasen-HHG pr¨asentieren wir hier auch eine vorl¨aufige Untersuchung der Kernelektronenabsorption f¨ur Aluminium in großen Mengen unter R¨ontgenstrahlung mittels time-dependent density functional theory (TDDFT). Ziel ist es hier zu ¨uberpr¨ufen, ob das unterstrichene theoretische Modell die bekannte Absorptionss¨attigung in Aluminium erfassen kann [8], um die Anwendbarkeit des Simulationsger¨usts f¨ur durch R¨ontgenpulse angetriebenes Festphasen-HHG abzusch¨atzen. Aus unseren Simulationen geht hervor, dass die Absorptionss¨attigung tats¨achlich reproduziert wird. Dies legt nahe, dass der erste Schritt von FestphasenHHG im dreistufigen Modell, n¨amlich die Anregung von Elektronen, durch auf TDDFT basierende ab initio -Simulationen erfasst werden k¨onnte. iv LIST OF FIGURES 5.2 Comparison of the transmission as a function of laser fluence between our theoretical calculation (blue curve) and the experimental data (purple curve) taken from the study [8]. Note that, since only the laser intensity is altered in the pulse and the laser fluence is proportional to the intensity, the axis of fluence can be seen as the axis of intensity. ........... 68 5.3 The absorption spectrum of aluminum near the LII,III absorption edge from theoretical calculation (red curve) and experimental data (blue curve) taken from the previous study [58]. The vector potential of the soft Xray used in this chapter is also shown as a sharp yellow peak to illustrate its position in the absorption spectrum. The full-width at halfmaximum (FWHM) of the laser is around 0.27 eV. Additionally, we also mark the energy difference of 19 eV between the absorption edge and the laser in the case of our simulation and the experiment. We select a soft X-ray with photon energy of 85.5 eV instead of 92 eV used in the experiment because it has the same distance from the edge in the absorption spectrum. ................................... 69 5.4 The decrease in energy as a function of fluence for the core level (blue curve) and a conduction band (red curve) compared to their original energy in the ground state. The arrow indicates the fluence around 5 J/cm2 where the core level has huge energy shift. Note that at this large energy shift in the core level the energy shift in the conduction band is negligible. This makes the photon energy of the soft X-ray off-resonance and thereby saturable absorption occurs. ......................... 70 5.5 Transmission as a function of laser fluence from theoretical calculation for the case with focal-spot average (red curve) and the case without focalspot average (blue curve). Transmission from the experimental data is also shown as the purple curve for comparison. ............... 72 5.6 Transmission as a function of laser fluence from theoretical calculation for the case with surface oxidation (red curve) and the case without surface oxidation (blue curve). Transmission from the experimental data is also shown as the purple curve for comparison. ................. 73 xii List of Abbreviations HHG high-order harmonic generation TDDFT time-dependent density-functional theory FWHM full-width at half-maximum FEL free-electron laser xiii List of Publications Chang-Ming Wang, Nicolas Tancogne-Dejean, Massimo Altarelli, Angel Rubio, and Shunsuke A. Sato, Role of electron scattering on the high-order harmonic generation from solids, [6], Physical Review Research, 2, 033333 (2020) xiv Chapter 1 Introduction Light enables us to see material world. Through light humans perceive and understand the world around us. At home, we see scenes via a window from the light transmitted through the material glass. In gatherings, we identify our friends by the light reflected from their faces. On a train, we read the information on a cell phone by the light radiated from the electronic screen. It is through light-matter interaction that we humans come to know the world. This concept is no exception in the realm of science, either. The very fundamental of scientific research is typically composed of four key steps: (1) observation of a phenomenon, (2) making a hypothesis to describe the observation, (3) verifying and reformulating the hypothesis to build up a theory, and (4) making predictions based on the theory. The very first step of all science, namely to see a phenomenon, is typically carried out by light-matter interaction. That is, scientists use light as a probe to see specific phenomena of interest. As a result, it is very important to understand the light-matter interaction so as to identify what are actually seen in the observation. And thereby one could establish solid ground truth for the following scientific investigation to carry on. This is the reason why optical and electron spectroscopy are among the most experimentally and theoretically studied topics, and many questions in the fields are still waiting to be answered. For a long time, physicists had set up a framework to describe light-matter interaction based on perturbation theory. In linear optics, the response of a dielectric material to light is simply formulated by electric permittivity. That is, the polarization of the material induced by the light is merely proportional to the electric field of the light by a scaling constant. To go beyond that, physicists also consider non-linear optics where the induced polarization is further expanded to some higher-order function of the electric field of the light. In either linear and non-linear optics, the light-matter interaction is always treated in the sense of a small perturbation: the influence of the light on the materials is assumed to be tiny. To be more precise, it is taken as given the electric field or intensity of the light is small compared to the electric field binding the electrons in the materials. Generally speaking, this is indeed true in many cases then and the lightmatter interaction is well understood and described by linear and non-linear optics. Nevertheless, such a successful perturbative approach eventually has to be revised and 1 1.1. HIGH-ORDER HARMONIC GENERATION reconsidered when the first laser was created in 1960 [9,10]. Unlike typical light, a laser is characterized by its narrow bandwidth and strong intensity, which renders in many cases the perturbative approach to light-matter interaction in linear and non-linear optics inapplicable. At the same time, with lasers being widely harnessed into experiments after its creation, many novel phenomena are observed and proposals for new applications are advocated. Because of these new observations and promising utilities, a new theory to describe the light-matter interaction is in great need in many scientific fields. One of the very interesting new subjects enabled by the creation of lasers in strong-field physics is the high-order harmonic generation (HHG). 1.1 High-order Harmonic Generation HHG is a photon up-conversion process at extreme occurring in a material when a sufficiently strong laser is applied. It is characterized by that the photon energy of the generated light is integer multiples, namely harmonics, of that of the input laser shining on the specimen. When such generated light is received by a detector and the intensity of different energy, or harmonics, is measured, it typically yields a power spectrum featuring strong peaks in the low-energy region and many weaker peaks with roughly equal intensity in the high-energy region, followed by a sharp drop in intensity at certain threshold energy (see Fig. 1.1 (a)). The region with weak peaks of similar intensity in the power spectrum is generally called the plateau of the spectrum and the end of this plateau the cutoff. The energy marking the cutoff is denoted as the cutoff energy and is a crucial property of a power spectrum of harmonic generations. People use the term HHG simply to indicate the high-energy photon emissions in the plateau region since they are of high interest in most of the cases. Such HHG was first observed in 1987 using rare gases as target specimens [11]. HHG involves light-matter interaction beyond the description of linear or perturbative non-linear optics due to the strong nonlinearity induced by the large intensity of the laser, and thereby several different approaches are proposed to describe how electrons in the target, typically gas molecules in a chamber at the time, respond to the strong laser. One of the most successful models describing the dynamics of the HHG using gas targets is the so-called three-step model [12,13]. In this semi-classical model the whole process of the HHG is explained by three distinctive steps: (1) An electron is ionized by the strong laser and escapes from its mother molecule. (2) The strong laser pushes the electron away from the ionized molecule and supplies it with energy. (3) When the electron is pushed back to the ionized mother molecule due to the oscillatory force of the laser, it recombines with the molecule and releases the extra kinetic energy gained during the second step by emitting a high-energy photon. These three steps are illustrated in Fig. 1.1 (b) as the process I, II, and II. This insightful semi-classical model provides an extremely simple physical picture for the HHG in gas while at the same time making several successful predictions on some important properties of a HHG power spectrum for atomic and molecular systems. For example, the cutoff energy of the spectrum could be estimated by the three-step model and its parabolic dependence on the electric field 2 1.1. HIGH-ORDER HARMONIC GENERATION II I III Position Energy Number of Photons Photon Energy Plateau Cutoff (a) (b) Figure 1.1: (a) A typical power spectrum for gas-phase HHG and (b) a schematic illustration of the three-step model. In (a), the spectrum shows several strong peaks in the low-energy region. In the high-energy region there are many weaker peaks with similar intensity, forming a plateau-like structure. The emission intensity drops quickly after a threshold defined as cutoff energy. In (b), the first, second, and third steps are marked by the process I, II, and III, respectively. The dark-blue circle describes a bounded electron and the light-blue one describes an energetic electron. The red curves represent the potential energy from the ion core and the laser. The orange sinusoidal curve is an analogy of a emitted photon. 3 1.1. HIGH-ORDER HARMONIC GENERATION amplitude of the driving laser could also be retrieved from the model. The understanding of the dynamics of the HHG brings forward several powerful applications based on these emitted high-energy photons. One of the most important applications for the HHG is the creation of ultra-short pulses [14,15,16]. Physicists have been utilizing the HHG photons from the plateau region to generate pulses with subfemtosecond duration [1,17], which is a typical time scale for the motion of electrons. This means one could use the ultra-short pulse synthesized from the HHG to observe electron motion in time and enhance our understanding of the corresponding processes. In practice, by utilizing ultra-short pulses researchers could perform real-time observation on electron transitions between atomic orbitals and electron tunneling through an atomic potential barrier with subfemtosecond temporal resolution [3,18]. In addition, similar techniques are also applied to reconstruct the highest-occupied molecular orbitals to understand the chemical properties of the molecules, and observe the attosecond dynamics of these orbitals, enabling the inspection of chemical reactions [2]. In short, the HHG from gas systems has become a new source of light and a deep understanding of its dynamics has enabled us to create ultra-short pulses from the HHG, which acts as a new light probe for matters and phenomena. With the gas-phase HHG the idea of ”seeing the matter through light” has advanced to a whole new level - into the world of electrons. Physicists now can utilize light beyond the description of linear or non-linear optics to see the material world in the natural length and time scale of an electron. The development of HHG is not limited to gas systems but advances to the world of condensed matters. In 2011 HHG from solid systems was discovered and this solid-phase HHG is observed to possess some similar properties like the gas-phase HHG [19,20,21]. For the radiation power spectrum, the solid-phase HHG also starts with high-intensity radiation peaks in the low-order region, follows a plateau region where peaks have lower and overall similar intensity, and then ends up at the cutoff energy. Likewise, the intensity of the solid-phase HHG cannot be described by perturbative non-linear optics. Considering the similarities, physicists have also been attempting to adapt techniques and applications for HHG from gas systems to solid systems. A successful extension would expand the field of strong-field physics and attosecond science into solid systems [22]. For instance, if the same technique for real-time observation of electronic transitions could be applied in solid systems, one would be able to obtain the dynamical involvement of each electronic bands and improve the understanding of the quantum interference between each excitation channel. In addition, the real-time observation of the HHG process in a solid system may also allow us to understand the role of other complicate mechanisms like scattering, which is actually one of the mechanisms we have studied in this thesis, in the making of solid-phase HHG. On the other hand, despite their similarity in radiation spectra for gasand solid-phase HHG, there are indeed several key differences between the two systems. In gas systems, the cutoff energy of a power spectrum has a quadratic dependence on the electric field amplitude of the laser while in solid systems it has linear dependence. Not only that, the cutoff energy in solid systems appears to be independent of the wavelength of the laser instead of inverse quadratic dependence in gas systems. Besides, unlike single plateau in a gas-phase HHG 4 1.1. HIGH-ORDER HARMONIC GENERATION spectrum, some theoretical simulations for solid-phase HHG show several plateau [23, 24,25,7], and a second plateau is observed in experiments using solids formed from rare gases [26]. Moreover, in most gas-phase HHG only electrons in the highest-occupied shell take part in the dynamics of photon emissions, while in solid-phase HHG the electrons lying in top-few valence bands could affect the radiation indirectly through quantum interference, leading to unconventional behavior like cancellation of radiation for certain polarization direction of the driving field [27]. All of these interesting properties and great prospects for applications call for the microscopic understanding of underlying dynamics and establishing a new model or theory for HHG from solid systems. Several theoretical approaches have been proposed to describe and reproduce solidphase HHG. Direct ab initio simulations using modeled crystal potentials reasonably capture the general structure of the HHG power spectra from solid systems [5,23,24]. Also, by performing ab initio simulations based on semiconductor Bloch equations [28, 29], one could obtain HHG power spectra directly comparable to experiments [30]. Besides reproducing numerical results with experiments, many simple semi-classical models are also proposed to describe the key essence of the dynamics. One of the very important models for solid-phase HHG is the three-step model generalized from gas systems. This generalized three-step model substitutes the parabolic energy dispersion in gas systems with complex band structures in solid systems [4,5]. In this model, the solid-phase HHG is characterized by the following three steps: (1) An electron is excited from the top of the top valence band to the bottom of the lowest conduction band, creating an electron-hole pair. (2) The created electron and hole are pushed by the external laser force and move in the conduction band and the valence band, respectively. (3) When the electron and the hole meet each other in real space, they recombine and emit a photon with energy equal to the energy difference between the conduction band and the valence band where they locate at that moment. A direct comparison of this generalized threestep model for solid-phase HHG and the original three-step model for gas-phase HHG is shown in Fig. 1.2. Although this generalized three-step model is limited to certain band structure of solid systems and intensity of the input laser, it does provide a simple description for the dynamics of HHG and connects the idea of HHG from gas and solid systems by a mere difference in the energy dispersion excited charged particles follow [31]. Despite the simple picture of solid-phase HHG by the generalized three-step model, the model still bears some non-intuitive assumptions that need to be revisited, quantified, and understood in details when one takes a closer look. Note that, within this semiclassical model the created electrons and holes are assumed to be effectively-free particles in the sense of effective mass. That is, they possess time-dependent effective mass and are only subjected to the force of the external laser. Such an assumption is suitable in the case of gas-phase HHG as the separation between an ionized electron and its parent ion is extremely large. So, the attractive force from the ion acting on a ionized electron could be neglected. However, in solids created electrons and holes are expected to still reside in the crystal and surrounded by a sea of positively-charged ions. In this situation, interaction with ions, like scattering between the charged particles and ions, 5 1.1. HIGH-ORDER HARMONIC GENERATION k Energy Gas-Phase HHG Solid-Phase HHG Ionization potential Band gap Atomic orbital Continuum Conduction band Valence band I II III I II III Figure 1.2: A schematic illustration for the HHG in gas (left) and solid (right) systems. The three steps for the two kind of HHG are marked by the procedure I, II, and II with different colors (red for gas and blue for solid). The energy diagram in the middle merges two kinds of energy dispersion of the gas and solid systems into one band structure. For the gas system, the energy of an atomic orbital is denoted by a horizontal line and that of the continuum is described by the parabola. For the solid system, the energy of the conduction band and the valence band are described by the parabola and inverse parabola, respectively. 6 1.1. HIGH-ORDER HARMONIC GENERATION Figure 1.3: Possible scattering of charged particles with ions. The trajectories of an electron(red curve) and a hole(blue curve) may pass ions (gray circles) several times and therefore scattering of the electron and hole with the lattice is generally expected. The trajectories shown in the figure ignore scattering and the electron and hole successfully recombine to emit a photon (orange sinusoidal curve). However, if the scattering is considered, the blue and red trajectory will change when making contact with the ions, the recombination of this electron-hole pair and the corresponding emission of a photon would become very unlikely. This indicates HHG could be altered by scattering between charged particles and the lattice. could be expected in general (see Fig. 1.3). As trajectories of electrons and holes in the second step will very likely be modified, the chances of recombination between them could become questionable. As a consequence, the corresponding photon emission and HHG might also be altered by the scattering. In addition, one would anticipate not just one but many scattering events when an electron or a hole is propagating in the lattice, because the charged particle generally travels several cells before recombination, indicating chances of multiple collisions with the lattice. Moreover, in real solid systems scattering is not only limited to the scattering between electrons and ions but also those between electrons and defects or phonons. Following the same argument that a trajectory could be modified by scattering, negligence of them impedes a proper description of the dynamics. As a result, it is important to understand the effect of and the role played by scattering for HHG in solid systems. In addition to the issue of scattering in solids mentioned above, we would also like to investigate how a solid-phase HHG power spectrum changes under an additional static field. For gas-phase HHG, physicists have been proposing to append a static electric field on the top of a main strong driving laser to enhance certain properties of the resulting spectrum. For example, theoretical simulations show a larger plateau region induced by a weak static field can be utilized to generate ultra-short pulses shorter than that 7 Chapter 2 Theoretical Models for the Quantum Systems The investigation on the effects of electron scattering and an extra static field in this thesis are made by performing quantum simulations and verify the physical consequences of the simulated HHG by semi-classical models. The quantum models are specifically chosen such that they are simple enough for us to focus on the effects under studies. On the other hand, they are at the same time complex enough to reasonably capture some behaviors of the HHG observed in experiments. Here, we will introduce the models for quantum simulations and describe how HHG power spectra are calculated. Two different quantum models are implemented for studying the effects of electron scattering and an extra static field. For the former case, we consider a one-dimensional non-interacting many-electron system formulated in the velocity gauge. First, the timeindependent Schr¨odinger’s equation with a given crystal model potential is solved to obtain its ground state. The state is then propagated over time under the influence of the external laser field to obtain its current density as a function of time. The HHG power spectrum is then calculated from the current density by performing Fourier transformation. As for the case investigating an extra static field, a simple parabolic two-band model is implemented [37]. This model is based on a three-dimensional non-interacting many-electron system in the velocity gauge and simplified by assuming parabolic valence and conduction bands and applying uniform matrix-element approximation. The system assumes an occupied valence band as the initial state and is propagated to calculate the current density at different time, which is then again Fourier transformed to obtain the corresponding HHG power spectrum. This much-simplified quantum model not only greatly reduces the computation cost but also ensures clear time-frequency analysis of the HHG signal, allowing for direct observation and investigation of the effect of an extra static field on spectra. There are two main sections in this chapter and they are dedicated to the two models mentioned above: In the first section, the Hamiltonian of the quantum model, time propagation of a state, and the calculation of the current density and thereby the HHG power spectrum are introduced. Additionally, we also explicitly show the numerical 15 2.1. QUANTUM MODEL implementation for each of the key steps in this model. In the second section, we introduce the parabolic two-band model and discuss the significance of the simplifications and how they are applied. The time propagation of a state and calculation of a HHG spectrum are also described shortly. 2.1 Quantum Model Here we present the quantum model for the study on the effect of Umklapp scattering to the HHG in solids. The model considered here is a one-dimensional, non-interacting, and many-electron system formulated in velocity gauge. We only consider a one-dimensional system here because in this study only linearly-polarized laser pulse is applied, which generally suggests that the major HHG signals also lies in one axis. However, the derivation in this section will still be presented in a general three-dimensional case so as to reuse the results here and simplify the derivation in the next section introducing the parabolic two-band model. In addition, we also assume electrons are independent with each others in the model. This independent-particle approximation was examined by the previous study based on TDDFT, in which almost the same HHG power spectra are reproduced with the independent-particle approximation, where the time-dependence of Kohn-Sham potential is ignored, describing the system as non-interacting particles in a mean-field potential [40]. The reduction of dimension and the neglect of interaction greatly reduce the cost of simulations. We also assume the laser field applied has no spatial variation, namely it is a homogeneous field, because the typical wavelength of the laser for generating HHG in solid systems is much larger than the length scale of a lattice cell. This assumption is commonly known as dipole approximation, and it enables us to reduce our simulation space to just one cell of the lattice, which again minimizes the computation cost. 2.1.1 Hamiltonian and the Ground State In a crystal, consider an N-electron system with Hamiltonian H[N]. The electronelectron Coulomb interaction is neglected so H[N]is simply a summation of Noneelectron Hamiltonian: H[N]= ΣN j=1H[j](2.1) with H[j]being the Hamiltonian of the jth electron in the N-electron Hilbert space H[j]=I⊗ · · · ⊗H⊗ · · · ⊗ I,j∈[1, N]. (2.2) ↑jth term Here, His the one-electron Hamiltonian in the velocity gauge H=(P−qA)2 2M+U(2.3) 16 2.1. QUANTUM MODEL where qand Mare the charge (sign included) and the mass of an electron and P=−iℏ∇ is the momentum operator. The vector potential of the external homogeneous laser field is designated as A=A(t) and the crystal potential is represented by U=U(r). The crystal potential has a translational symmetry with a primitive lattice vector L, namely U(r+L) = U(r). From the fact that the Hamiltonian of this N-electron system is merely a summation of None-particle Hamiltonian (namely, electrons are not coupled) and the observable we are interested in is just a single-particle operator, we can describe the whole system with just one single Slater determinant. That is, the electrons in the system could be dealt with one by one instead of all together at the same time. In other words, this N-electron system can be effectively reduced to None-electron systems. Therefore, we will consider a one-electron system with Hamiltonian Hfrom here on. The calculation of energy eigenvalue problem for the ground state can be simplified by applying Bloch’s theorem. According to the theorem, an energy eigenstate ψ(r) = ⟨r|ψ⟩ in a crystal could be described by the form ψ(r) = ψk(r) = uk(r)eik·r(2.4) with uk(r) being the Bloch function with a translational symmetry of Land kbeing the crystal momentum. Now let ⟨r|ψnk⟩=⟨r|nk⟩=ψnk(r) = unk(r)eik·rbe an energy eigenstate with the band index nand the crystal momentum kand its corresponding energy be Enk. By applying |nk⟩and ⟨r|on the right and left of Eq. 2.3, we have: ⟨r|H|nk⟩=eik·r Mh−ℏ2 2∇2−iℏ(ℏk−qA)· ∇ +|ℏk|2 2+|qA|2 2−ℏqk·Aiunk(r) + eik·rU(r)unk(r) =Enkeik·runk(r) or, after canceling the same terms for the two equations on the right: Enkunk(r) = h−ℏ2 2M∇2−iℏ M(ℏk−qA)· ∇ +|ℏk|2 2M+|qA|2 2M−ℏq Mk·A+U(r)iunk(r) =h(−iℏ∇+ℏk−qA)2 2M+Uiunk(r) =H[k]unk(r). (2.5) This eigenvalue problem without external laser, namely A(t) = 0, is then solved numerically for each kto form the ground state of the system. In this model, we will consider an N-electron ground state with fully occupied valence bands and completely empty conduction bands. This ground state also serves as the initial state of the manyelectron system. The corresponding single-electron state at crystal momentum klying in a valence band νat the beginning of the time t= 0 will be designated by ϕνk(r;t= 0) = ⟨r|ϕνk(t= 0)⟩=⟨r|ψνk⟩. (2.6) This state will be propagated over time for the calculation of HHG power spectra. 17 2.1. QUANTUM MODEL 2.1.2 Time Propagation and HHG Power Spectrum In this model, the time evolution of the single-electron state |ϕ(t)⟩is governed by Schr¨odinger’s equation: iℏ∂ ∂tϕνk(r;t) = ⟨r|H|ϕνk(t)⟩(2.7) or, after applying Bloch’s theorem with ϕνk(r;t) = uk(r;t)eik·r, we have iℏ∂ ∂tuk(r;t) =(−iℏ∇+ℏk−qA(t))2 2Muk(r;t) + U(r)uk(r;t). (2.8) Note that, at the beginning of the time the Bloch function is just the one of a valence band specified by ν, namely uk(r;t= 0) = uνk(r). Eq. 2.8 will be the actual dynamical equation implemented in our numerical simulations. Once we have the state at any given time, the current density J(t) could then be calculated by J(t) = X ν 1 ΩBZΩB Jνk(t)dk(2.9) Jνk(t) = q M⟨ϕνk(t)|P−qA(t)|ϕνk(t)⟩ =q M 1 ΩCZVC dru∗ νk(r;t)(−iℏ∇+ℏk−qA)uνk(r;t) where ΩBand ΩCare the volume of the first Brillouin zone and the volume of a primitive cell. And the RΩBdkand RΩCdrindicate integration over the first Brillouin zone and one primitive cell, respectively. Note that Jνk(t) is the contribution from one electron to the total current density J(t) of the N-electron system. Here we would like to address that νand kin the above equations merely indicate what the band index and crystal momentum a state ϕνk(t) have at the initial time and they do not suggest the corresponding state is always in the valence band denoted by νover time. As the electric field of the light emission is proportional to the acceleration of charged particles, we could extract the HHG power spectrum through the Fourier transformation of the current. Therefore, the emission power spectrum P(ω) can be calculated by P(ω) = ω2|FT {J(t)W(t)}(ω)|2(2.10) with FT being Fourier transformation and W(t) a window function. The application of a window function onto current density before Fourier transformation allows for a smoother spectrum. Here we will simply use the envelop of the laser field as the window function. An example for the HHG power spectrum is given in Fig 2.1, together with the electric field of the corresponding driving laser and the current density the laser creates. 18 2.1. QUANTUM MODEL Figure 2.1: Some examples of numerical calculations for HHG in a solid system based on the one-dimensional quantum model introduced in this section. The electric field of an applied laser as a function of time is described in (a). The current density generated from the system as a response to the applied laser field shown in (a) is also described in (b) as a function of time. The HHG power spectrum calculated from the current density in (b) is shown in (c). 2.1.3 Time-Frequency Analysis To study the dynamics of the HHG, we will perform time-frequency analysis to extract the emission time and energy of the generated photons. This analysis involves superimposing the predictions from a semi-classical model onto the results from a quantum simulation. Therefore, one can trace the origin of the emission and understand how they are affected by scattering. The details of the semi-classical models will be described in the next chapter. For the quantum signals, we perform the short-time Fourier transformation, specifically the Gabor transform, to the current density to obtain its emission strength as a function of emission time τand emission energy (or equivalently photon frequency ω): G(τ, ω) = FT {J(t)e −(t−τ)2 2σ2}(ω) 2(2.11) with σbeing the standard deviation of the Gaussian. Note that this Gabor transformation is nothing but Fourier transform of the current density sampled by Gaussian function near a specified time τ, and therefore one can interpret the result G(τ, ω) as the emission spectrum from the system around time τ. How closely this spectrum is related to the emission time τis determined by the standard deviation σ. Small σsuggests the Gaussian window vanishes very quickly with time tmoving away from τand 19 2.1. QUANTUM MODEL thereby current density at time far away from τwill take little part in the final spectrum G(τ, ω). Note that the value of this σshould be smaller than the time scale of an emission event, otherwise such events can not be resolved in the time domain and the information for the time of recombination is lost. As an emission event, which consists of the motion of an electron moving away from and coming back to a core ion, is mainly controlled by the driving laser, we can roughly estimate the time scale of an emission event to be one optical cycle of the laser. Therefore, σis always smaller than one optical cycle. However, σshould not be too small so as to avoid large uncertainty in frequency domain (or equivalently, energy domain). In practice, one would typically tune σto make sure the signal in the energy range of interest has good resolution in both time and frequency. The short-time Fourier transformation allows us to extract the emission time τand emission energy ℏωof HHG from the quantum system. On the other hand, we can also calculate the timing and energy from a semi-classical model. When the two signals are plotted together and match each other to a reasonable degree, one could then attribute the origin of a photon emission to a specific classical trajectory in the semiclassical model. This procedure is commonly addressed as time-frequency analysis. If an HHG event is successfully characterized to a trajectory in the procedure, this analysis could provide crucial information for the HHG like time of excitation, motion of the corresponding electron-hole pair, and interaction between the pair and the external laser. Therefore, time-frequency analysis could allow for a deeper understanding of the quantum signal and dynamics of HHG. In Fig. 2.2, we demonstrate a simple example of the time-frequency analysis. The short-time Fourier transformation of the current density from a quantum simulation G(τ, ω) is plotted as the color map in the background of the figure, and the classical predictions are marked by black circles. The classical predictions from a semi-classical model will be introduced and connected to the time-frequency analysis here in the next chapter. One can see that, in the energy domain of 7 to 14 eV and time domain of 35 to 65, the structure of the major time-frequency signals of G(τ, ω) (colored in red) are reasonably reproduced by the classical results. We can than connect this major emission with the specific trajectory and investigate its dynamics. 2.1.4 Numerical Implementation Here we will describe the numerical implementation on solving the energy eigenvalue problem in Eq. 2.5 for the ground state and propagating the wavefunction of an electron over time according to Eq. 2.8 for the current density. As we are actually dealing with a one-dimensional system, the corresponding Hamiltonian, dynamical equation, and other quantities shown here will be reduced to their one-dimensional versions, e.g., k→k, r→r,A(t)→A(t), L→Letc. Evenly-distributed grids are employed to discretize both real-space (r-space) and 20 2.1. QUANTUM MODEL Figure 2.2: An example for the time-frequency analysis. The spectrum of the current density from the short-time Fourier transformation is shown as the color map in the background, and the emission timing and energy from a semi-classical model is marked by black circles. crystal-momentum space (k-space). For r-space, we have: rj= ∆r×(j−0.5) −L 2,j∈ {1,2,· · · , Nr}(2.12) where Nris the number of grid points and ∆r=L Nris the grid spacing. Note that the choice of shift here guarantee inversion symmetry for the grids. For k-space, the grid points are also defined in the similar way as: kj= ∆k×(j−0.5) −π L,j∈ {1,2,· · · , Nk}(2.13) where Nkand ∆k=2π LNkare the number of grid points and the grid spacing, respectively. The numerical calculation of derivatives is carried out simply by finite difference method to the 4th-order (using 5 grid points). The first and second derivatives of the Bloch function unk(r) with respect to rat r=rjcan be calculated by (let uj=unk(rj) for simplicity): ∂unk(r) ∂r r=rj ≈X l=−2,··· ,2 D[1] luj+l ∆r,(D[1] 0, D[1] ±1, D[1] ±2) = (0,±2 3,∓1 12) (2.14) ∂2unk(r) ∂r2r=rj ≈X l=−2,··· ,2 D[2] luj+l ∆r2,(D[1] 0, D[1] ±1, D[1] ±2) = (5 2,4 3,−1 12). (2.15) 21 2.1. QUANTUM MODEL Keep in mind that the space is periodic so the index j=N+1 and j=N+2 correspond to j= 1 and j= 2 and j= 0 and j=−1 correspond to j=Nand j=N−1. With the numerical form of the derivatives, the one-dimensional version of Eq. 2.5 can be written in the form Enkuα=Pβhαβuβas shown by the following: Enkuα=H[k]uα =−ℏ2 2MX γ D[2] γ ∆r2uα+γ−iℏ M(ℏk−qA)X γ D[1] γ ∆ruα+γ+|ℏk|2 2M+|qA|2 2M−ℏq MkA +U(rα)uα =X βn−ℏ2 2MX γ D[2] γ ∆r2δβ,α+γ−iℏ M(ℏk−qA)X γ D[1] γ ∆rδβ,α+γ +δαβ|ℏk|2 2M+|qA|2 2M−ℏq MkA +U(rα)ouβ =X β hαβuβ(2.16) with γan integer running from −2 to 2. After setting A= 0, the eigenvalues and eigenfunctions of the matrix hwill then be solve with conventional numerical solver. In the time domain, the time axis is discretized evenly with a time step ∆tfrom t= 0 to a specified time. A Bloch state at the next time grid is calculate through the application of time propagator on the corresponding state at present time grid according to the one-dimensional version of Eq. 2.8. Therefore, we have uk(t+∆t) = eH[k](t) iℏ∆tuk(t). For the time propagator, Taylor expansion to the 4th order is implemented so the Bloch state at next time is numerically calculated by: uk(t+ ∆t)≈1 + X j=1,··· ,4 1 j!((H[k](t))∆t iℏ)juk(t). (2.17) The actual operation for applying multiple H[k]on the Bloch state can be done by utilizing the right-hand side of Eq. 2.16 repeatedly. Finally, the evaluation of the current density J(t) at any given time grid is numerically carried out by the discretized form of one-dimensional version of Eq. 2.9: J(t) = 1 ΩBX ν,α Jνkα∆k(2.18) Jνkα=q MΩCX βh(ℏk−qA(t))|uνkα(rβ;t)|2∆r−iℏX γ=−2,··· ,+2 u∗ νkα(rβ;t)D[1] γuνkα(rβ+γ;t)i. To have reliable numerical results, simulations are verified by gradually using smaller ∆r, ∆k, and ∆tand observe the converging behavior for the current density and the corresponding HHG spectrum. 22 2.2. THE PARABOLIC TWO-BAND MODEL 2.2 The Parabolic Two-Band Model The addition of a static electric field to the laser invokes various changes in electron dynamic, altering the corresponding HHG in a very complex and intertwined way. Therefore, we would like to really simplify the quantum model so as to focus on the mechanism of interest and understand the impact of the static field. The specific mechanism we would like to investigate here is how the change of the trajectory of an electron by the extra static field makes its way to the corresponding HHG power spectrum. To achieve this, we consider a non-interacting, many-electron model like the one introduced in the previous section but reduce the number of bands to just one conduction band and one valence band. Additionally, we avoid solving the energy structure of the system and just explicitly assign parabolic bands to the model. After these major reductions, various minor approximations are made to further reduce the complexity of the system and facilitate simulations of the model. 2.2.1 The Dynamical Equation and Approximations Here we first introduce the Houston’s state [41] as they will be utilized as a basis set to expand the dynamical equation in Eq. 2.8 for a Bloch wavefunction. The Houston state ηnk(r;t) could be define as the instantaneous energy eigenstates at time twith a phase accumulated over its trajectory K(t) in crystal-momentum space [41]: ηnk(r;t) = eRt 0 EnK(t′) iℏdt′unK(t)(r) (2.19) where unK(t)(r) is the instantaneous energy eigenstate at time twith EnK(t)being the corresponding energy: EnK(t)unK(t)(r) = H[K(t)]unK(t)(r). (2.20) The trajectory in crystal-momentum space K(t) here can be calculated by the acceleration theorem: K(t) = k−q ℏA(t) (2.21) where kis the initial crystal momentum of the state. Notice that, by performing time derivative on Eq. 2.21, one could obtain the Newton-like equation: dℏK(t) dt =qE(t) = Flaser(t) (2.22) with Flaser being the force of the laser exerted on an electron. The laser applied within this model will be composed of an oscillating field and a static field. The oscillating part is a typical linearly-polarized laser. The electric fields from both components are assumed to lie in the zdirection. 23 3.1. SEMI-CLASSICAL MODEL FOR THE HHG IN GASES AND SOLIDS stationary compared to the motion of the ionized electrons during the whole process of the HHG due to its much-larger mass. In the second step, the ionized electron now is driven by the external laser and move in place far from the ionized parent molecule. The motion of the electron at this stage can be accurately captured by classical mechanics simply with a free electron accelerated by the laser. The electron gains energy from the laser and accumulates it in terms of its kinetic energy. Finally, in the third step, the force of the oscillating laser changes its direction and pushes the electron back to the ionized parent molecule. If the electron recombines with the parent ion, it releases its high kinetic energy by emitting a high-energy photon. This completes one cycle of the HHG in a gas system. The most important consequence of the three-step model is that it gives the whole process of the gas-phase HHG a very simple physical picture. One could intuitively estimate the change of the HHG when the laser is altered by the classical mechanics in the second step instead of complicate quantum dynamics. Furthermore, by simulating the classical trajectories of the model and compare the signals from experiments or quantum simulations, one could analyze the source of the HHG and distinguish the major contributor of the process. This analysis is then the time-frequency analysis which we have already discussed about in Chapter 2.1.3. In addition to the simple picture, the model also allows for prediction of the cutoff energy Ucof the corresponding power spectrum: Uc=Ip+ 3.17Up(3.1) where Ipis the ionization potential and Up=q2F2 0 4Mω2 0is the ponderomotive energy with F0and ω0the electric field amplitude and angular frequency of the laser, respectively. Although the actual value of the cutoff energy could deviate from Eq. 3.1, its quadratic dependence on electric field amplitude (through Up) is indeed in accordance with experiments for gas-phase targets. This ability to estimate the cutoff energy is crucial for developing applications based on the HHG. The very success of the three-step model in gas systems inspires the adaption of the model to the HHG in solid systems. And one of the very important adaptation is made by Vampa et al. through the incorporation of the band structure[4]. In the second step of the three-step model, the motion of a free electron driven by a laser can be equivalently treated as an electron resides in a parabolic band and move under the influence of the laser. For solid-phase HHG, the idea of this parabolic band is extended to the actual band structure of the solid system. The generalized three-step model describes the HHG in solid systems by the following three characteristic steps:  Excitation of an electron from a valence band to a conduction band, creating an electron-hole pair.  Acceleration of the electron-hole pair by the laser force, gaining energy prescribed by the band structure.  Recombination of the electron-hole pair, emitting a photon with energy of the band energy difference. 30 3.1. SEMI-CLASSICAL MODEL FOR THE HHG IN GASES AND SOLIDS Figure 3.1: A schematic illustration for the generalized three-step model. The blue and red curves describe a valence band and a conduction band. The steps of excitation, acceleration, and recombination are marked by I, II and III, respectively. These three steps are illustrated schematically in Fig. 3.1. For the first step, laser excites an electron from the top of a valence band to the bottom of a conduction band and an electron-hole pair is created, as shown by the process I in Fig. 3.1. In the real space, the electron and the hole are assumed to be in the same position at this instance. In the second step (process II in Fig. 3.1), the electron and the hole will be driven by the laser and move in k-space according to the acceleration theorem with their crystal momentum over time being K(t). In the real space, the electron and the hole also propagate with their instantaneous velocity specified by the corresponding group velocity vg n(k) = 1 ℏ∇kEnk, (n∈ {c, ν}). For the final step, during the motion of the electron and the hole, they are considered to collide with each other and recombine if they move to the same position in real space. The electron in the conduction band then drops down to the valence band as shown in process III of Fig. 3.1 and releases its extra energy, namely the energy difference between the two bands, by creating a photon of that energy. These three steps complete a cycle of photon emission for solid-phase HHG. Now we will explicitly show the procedures for implementing the three-step model for solid-phase HHG. The description here will be limited to a one-dimensional system since the semi-classical model for both of the quantum models described in the previous chapter effectively reduces to such a one-dimensional model. In addition, we also consider only the solid systems of direct band gap with its band gap locates at Γ point (k= 0) and has zero group velocity at this point. We should first define the physical quantities involved in the calculation. After an electron is excited by a laser, or equivalently an electron-hole pair is created, at time tex, we can calculate the relative position between the electron and the hole 31 3.2. SEMI-CLASSICAL MODEL WITH UMKLAPP SCATTERING x(t) = xe(t)−xh(t) where xe(t) and xh(t) are the displacement of the electron and the hole, respectively. Note that the relative velocity of the electron and the hole v(t) = dx(t) dt is equivalent to their corresponding relative group velocity, namely v(t;tex) = vg c(K(t;tex)) −vg ν(K(t;tex)) where K(t;tex) is the displacement in crystal momentum for the electron-hole pair created at time tex: K(t;tex) = −q ℏ[A(t)−A(tex)] , t ≥tex. (3.2) The calculation of the semi-classical model is carried out by the following procedures: 1. Choose an excitation time tex in the duration of the laser pulse. 2. Set up the initial condition in real space: x(t=eex) = 0 and v(t=tex) = 0. 3. Calculate the trajectory in the real space by one time step according to x(t) = Rt tex v(t′;tex)dt′. 4. If x(tr) = 0 at certain time tr, a recombination has occurred. The time is then designated as recombination time tr. 5. Calculate the energy of emitted photon ℏωfrom this recombination by: ℏω=EcK(tr;tex)−EνK(tr;tex). (3.3) 6. Go back to step 3 and repeat for x(t) at another time step until reaching the end of the pulse. 7. Go back to the first step and repeat with a new excitation time t′ ex until the excitation time reaches the end of the pulse. The numerical calculation of x(t) is carried out by Runge-Kutta method to the 4th order with time step ∆tRK. Also, the deterministic condition x(tr) = 0 is substituted by the condition x(tr−∆tRK)×x(tr+ ∆tRK)≤0 because absolute zero is unattainable in the numerical calculation. Note that with the substituted condition the program may fail to detect a recombination when the trajectory x(t) just touches the time axis in the x−tplane without passing it. This kind of scenario could be reduced by using a smaller ∆tRK. As a result, when checking the convergence of the time step, one should make sure not only the trajectory x(t) = Rt tex v(t′;tex)dt′but also the condition x(tr) = 0 reach convergence. 3.2 Semi-classical Model with Umklapp Scattering Despite the direct analogy between the three-step model and the generalized one from gas systems to solid systems, there are many mechanisms not included in the generalized three-step model. One of them is the scattering between electrons and the lattice (sea of positively-charged ions). In the second step of the generalized three-step model, an 32 3.2. SEMI-CLASSICAL MODEL WITH UMKLAPP SCATTERING electron is treated as an effectively-free electron driven by a laser and moves in the energy landscape, namely energy dispersion, formed according to the lattice potential. That is, a parabolic energy dispersion meant for a free electron is distorted to a complicate band structure to reflect the energy structure created by the lattice force. And this energy structure is taken in the semi-classical model to account for the force of the lattice implicitly so as to treat the electron as an effectively-free particle only subjected to the force of the external laser. As a result, this effectively-free electron picture in the second step of the generalized three-step model ignores any scattering with the lattice. The neglect of such scattering could be questionable as it might make a difference for the solid-phase HHG. Under a typically setting for generating high-order photons in a solid system, electron-hole pairs responsible for the light emission would generally travel a distance of several lattice cells. Therefore, one would expect the electron-hole pairs have chances to scatter with some ions during the process. If such scattering does occur, the scattered electron-hole pair should have quite different speed and direction compared to those before the scattering. This suggests the trajectory of the electron-hole pair would be altered to a great extent, leading to a modified chance of recombination and photon emission in the future. As a consequence, it is natural to think the HHG power spectrum would be altered by the scattering in a crystal. The scattering could be integrated into the generalized three-step model by a small modification in the second step. We will demonstrate this here by first looking at the scattering process in the k-space. In principle, as an electron passes a point, which is addressed as a scattering point in our context, in a band where a neighboring band has very similar or even the same energy, the electron would have chances to move to this neighboring band by shifting its crystal momentum by a reciprocal lattice vector. This process is typically known as Bragg scattering or Umklapp scattering. In Fig. 3.2, such a process is illustrated with a conduction band (solid curve) and the shifted ones (dotted curves) of a one-dimensional system. Note that the scattering points for this specific band structure is found at the cross points of the band and the shifted ones. Before the scattering, an electron driven by a laser happens to be moving toward a scattering point as indicated by the process I in Fig. 3.2. After passing the scattering point, there are two possible paths: the non-scattered path and the scattered path, which are represented by the process II and II’ in Fig. 3.2, respectively. The idea and the effect of the scattering would become more clear when we observe the two different paths in the real-space. For path II, the velocity of the electron will change continuously with time so the corresponding trajectory xe(t) will be a smooth curve without abrupt bending. On the other hand, for path II’ the discontinuous group velocity of the electron when going from the conduction band to the shifted conduction band leads to a abrupt bending in xe(t) at the time of scattering. The trajectory will not look smooth due to this scattering. The behavior of the electron of the path II’ then looks like a classical particle hitting an obstacle and scattered off. This reestablishes the familiar idea of scattering. From this example in one-dimensional system, it is clear that we need to consider not only the conventional non-scattered path but also the scattered path in order to take account for the effect of Umklapp scattering. Therefore, we can incorporate Umklapp scattering into 33 3.2. SEMI-CLASSICAL MODEL WITH UMKLAPP SCATTERING Figure 3.2: A sketch for the two possible paths for an electron which undergoes Umklapp scattering in the crystal-momentum space. The process I indicates an electron moving towards the scattering point located at zone boundary. The process II and II’ describe the non-scattered and scattered paths, respectively. the generalized three-step model by creating a scattered trajectory whenever a scattering point is traveled through by an electron-hole pair. In other words, in the second step of the generalized three-step model, a trajectory will branch into two trajectories, one for the non-scattered and the other for the scattered, at the time of scattering. In summary, our semi-classical model with Umklapp scattering describes the HHG in solid systems by the following three steps:  Excitation of an electron from a valence band to a conduction band, creating an electron-hole pair.  Acceleration of the electron-hole pair by the laser force, gaining energy prescribed by the band structure. The trajectory of the pair branches into one scattered trajectory and one non-scattered trajectory if a scattering point is passed.  Recombination of the electron-hole pair, emitting a photon with energy of the band energy difference. Note that step one and three remain exactly the same as those in the generalized threestep model, and step two is modified by the addition of trajectory branching. We will see in Chapter 4our model can reproduce the quantum result and describes the structure of multiple plateaus, which is unavailable in the three-step model without scattering. 34 3.2. SEMI-CLASSICAL MODEL WITH UMKLAPP SCATTERING 3.2.1 Implementation of the Model with Umklapp Scattering The implementation of the semi-classical model with Umklapp scattering deserves a bit more attentions because of the possible exponential growth of branching trajectories. Notice that the number of trajectories is doubled every time an electron passes through a scattering point and it depends sensitively on the setup of the laser pulse. For example, increasing laser intensity would lead to larger span in the crystal-momentum space for an electron-hole pair and enable the pair to reach more scattering points. Additionally, using a longer pulse with more optical cycle will also increase the repetition of an electronhole pair passing a scattering point. Due to the large number of trajectories from the complex scattering process, it would not be computationally efficient to fully simulate the new semi-classical model to the end of the pulse. A simple solution is to stop branching or just terminate the simulation of a trajectory once it becomes less important. For example, we can disregard scattered trajectories if the current trajectory has already undergone Umklapp scattering for several times, because in general a multi-scattered electron-hole pair are very unlikely to meet and recombine in the future. In practice, this is carried out by stopping branching at scattering points for doubly-scattered trajectories. Also, we can neglect the photon emission beyond first recombination due to the nature of quantum spreading of the electron-hole wave packets in real-space. This allows us to terminate a trajectory at the time of first recombination. The implementation of the model could be listed in the following steps: 1. Choose an excitation time tex in the duration of the laser pulse. 2. Set up the initial condition in real space: x(t=tex) = 0 and v(t=tex) = 0. 3. Calculate the trajectory in the real space by one time step according to x(t) = Rt tex v(t′;tex)dt′. 4. In each time step, designate a recombination event and recombination time t=tr if the condition x(t) = 0 is satisfied. Then:  Calculate the energy ℏωof emitted photon by ℏω=EcK(tr;tex)−EνK(tr;tex).  Jump to the last step. 5. In each time step, designate a scattering event and scattering time tsif the condition K(t;tex)−ks= 0, where ksis one of the scattering points, is satisfied. Then:  Create a scattered trajectory if the total number of scattering is no more than two.  Stay with the current trajectory if the total number of scattering is larger than two. 6. Go back to step 3 and repeat for x(t) at another time step until reaching the end of the pulse. 35 3.2. SEMI-CLASSICAL MODEL WITH UMKLAPP SCATTERING 7. Go back to the first step and repeat with a new excitation time t′ ex until the excitation time reaches the end of the pulse. For the same reason in the previous section, the deterministic condition K(t;tex)−ks= 0 is substituted by the condition [K(t−∆tRK;tex)−ks]×[K(t+ ∆tRK;tex)−ks]≤0. Again, this substituted condition is sensitive to the time step ∆tRK and its convergence should be verified. 36 Chapter 4 The Effects of Scattering and a Static Field on the HHG in Solids We study the dynamics of solid-phase HHG with the effects of scattering and a static field through the comparison between quantum and semi-classical models. As data from experiments or ab initio quantum simulations is generally a result of intricate dynamics, it is very difficult to delineate the mechanisms involved. In this study, to clarify a connection between the HHG in the energy range of interest and the external laser, we employ simplified quantum models, namely the one-dimensional quantum model and the parabolic two-band model introduced in Chapter 2. Furthermore, in order to develop microscopic insight into the phenomena, we follow the conventional approach in the community of HHG studies, namely a comparison between quantum and semi-classical models. To be more precise, we will use the semi-classical models to retrace the HHG signal from quantum simulations and establish its characteristics in the aspect of the scattering and a static field. In the investigation of scattering effects, we begin by briefly looking at the different dependence of the cutoff energy on the electric field amplitude of the external laser between a parabolic band and a non-parabolic band, which is the Kane’s band in our case, within the generalized three-step model. The empirical formula of the cutoff for the non-parabolic band is also proposed, which could serve as a quick estimation for experiments on solids with band structure in the form of the Kane’s band. With a basic understanding of the behavior of the cutoff, we then move to the main investigation. Quantum simulations based on the one-dimensional quantum model using a model potential formulated in Sec. 2.1 are performed and their quantum data is compared with the predictions from the generalized three-step model with Umklapp scattering described in Sec. 3.2. Multiple plateaus are observed in quantum simulations, and the semi-classical model successfully captures the cutoff energy for each plateau. In addition, with the help of the time-frequency analysis, we show that the major contribution to each plateau comes from the recombination of electron-hole pairs undergo a different number of scattering. Finally, we present a semi-classical approach extended by using the concept of the mean-free path to incorporate the general scattering effects in solid. 38 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG From this semi-classical model, the wavelength independence of the cutoff energy in solid-phase HHG could be reproduced. For the effect of a static field, we use the parabolic two-band model derived in Sec. 2.2 and compare the results with the predictions from the generalized three-step model in Sec. 3.1. It is observed that the cutoff energy of the HHG power spectrum splits into two by the static field. The two cutoffs depend on the static electric field amplitude in the opposite way: one increases and the other decreases with the increasing static field. However, the increasing cutoff will saturate at maximum energy at a certain electric field amplitude and starts to decrease with the stronger static field. From the semiclassical model, it could be shown that the splitting of the cutoff energy is a result of broken symmetry induced by the static field, which removes the two-fold degeneracy of trajectories for the electron-hole pairs. This chapter consists of two major parts corresponding to the two subjects mentioned above. In section 4.1, we present our studies on the effect of Umklapp scattering on the solid-phase HHG. Additionally, the alternative approach to general scattering effects in solids is also presented at the end of this section. In section 4.2, the effect of a static field on solid-phase HHG are demonstrated and discussed. In both of the two sections, the setup of the related models and numerical parameters are also given at the start of each section. 4.1 The Effect of Umklapp Scattering on Solid HHG 4.1.1 Setup of the Quantum System under a Model Potential We would specify the actual numerical setup of the quantum system for studying scattering effects and briefly discuss the substitution of the band structure in the semi-classical calculation. In quantum simulations, the model potential V(x) of the Matthieu-type in the following form are utilized: V(x) = V0cos(2π Lx) (4.1) with V0= 0.37 a.u. and L= 8 a.u. These specific values are taken from the previous study [7] since we will make several comparisons with their results. The corresponding band structure is computed by diagonalizing the Hamiltonian and yields a band gap of ϵg= 4.18 eV and reduced electron-hole effective mass of µ= 0.083 electron mass. The band structure is shown in Fig. 4.1 (a). For simplicity, in the calculation of semi-classical models, this numerically solved band structure is substituted by the Kane’s band to reduce the complication induced by tunneling between bands. Notice that, in principle, one can employ the numerically evaluated band structure shown in Fig. 4.1 (a), by taking account of the quantum transitions between different bands with a non-zero energy gap. To comply with the definition of non-scattered trajectories discussed in Chapter 3and neglect the complication in the states due to tunneling processes, we substitute the band structure solved numerically from the potential in Eq. 4.1 with the Kane’s band, which always has zero energy gap between the current band and shifted bands. As a result, 39 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG Figure 4.4: The color map shows the calculated HHG power spectra from the onedimensional quantum simulations versus the electric field amplitude F0of the laser. The cutoff energy computed by our semi-classical model using the Kane’s band is described as different lines: The cutoff from trajectories without scattering is shown as the black-solid line. The cutoff from trajectories with single scattering is described as the blue-dashed line. The cutoff from trajectories with double scattering is shown by the red-dotted line. This figure is taken from our publication [6]. 46 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG Figure 4.5: Time-frequency analysis of the HHG calculated from quantum simulations are shown as the color map. The emission energy and timing predicted from the generalized three-step model with Umklapp scattering are represented as black dots, and their contribution for no scattering, single scattering, and double scattering are indicated in panel (a), (b), and (c) respectively. This figure is taken from our publication [6]. 47 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG emissions at any time. On the other hand, in our model which considers both realand k-space trajectories, electrons and holes are treated like point particles(local). The recombination of an electron-hole pair is only allowed when they are right at the same place. And therefore photon emission only occurs at a certain time. Generally speaking, electrons and holes in the solids are not point particles localized to a point in both real and k space but are wave packets with finite spreading. The assumption of point particles in our model gives rise to the deviation between the quantum signal and the semi-classical predictions in Fig. 4.5. For example, the semi-classical prediction fails to capture the signals on the left side (around 40 to 50 fs) of the semi-classical predictions in the second plateau (around 15 to 35 eV). We could further assess how local an electron-hole pair actually is by including the spreading of their wave-packets into the semi-classical model. This would allow for a quantitative measure of the spreading and verification of the assumption of point particles in the model. In addition to the examination of wave-packet spreading in real space, it is also worth investigating the spreading in k space. Within the semi-classical model it is assumed electrons are excited and tunnel to a conduction band only at the Γ point. This may not be realistic as, in principle, electrons in the region near the point could also become excited and tunnel to a conduction band. That is, electronic excitation is in general not localized at the Γ point but has a spreading near the point. As a quick summary, the finite spreading of the electron-hole wave packet in real space affects the constraint of the recombination for the pair, and the finite spreading of the tunneling region near the Γ point affects the excitation channels of the electron-hole pair. The generalization of the semi-classical model for the spreading of electron-hole pairs in real and k space is straight-forward: For the spreading in real space, the condition for recombination in the semi-classical model is modified such that electrons and holes are allowed to recombine as long as they are within a specified distance |∆x|instead of at exactly zero distance. As for the spreading in k space, the initial crystal momentum of an electron-hole pair now is not limited to the single Γ point but to a small region centered at the Γ point with a specified radius |∆k|. Beyond these modifications in each case, the semi-classical model remains unchanged. In Fig. 4.6 the results from the semi-classical model with different constraints for recombination are shown: Recombination of an electron and a hole is allowed (a) only when they are at the same place in real space and (b) when their separation distance in real space is within 40 a.u. We could see that the relaxation for recombination constraint in the real space does not improve the semi-classical prediction qualitatively. This suggests the assumption that electrons and holes are like point particles works well in real space, at least for this specific quantum system. In Fig. 4.7, the semiclassical predictions for different excitation channels near Γ point are shown: In panel (a) excitation occurs exactly at the Γ point. In panel (b) excitation occurs within a radius of 0.1 a.u. centered at the Γ point. It is clear that the missing predictions on the left of semi-classical results in the second plateau mentioned earlier now appear. This indicates that non-zero spreading of excitation channels for electron-hole pairs in 48 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG Figure 4.6: The comparison for semi-classical predictions for recombination occurring (a) at zero separation distance and (b) within a separation distance of 40 a.u. between an electron and a hole. The purple circles describe the semi-classical predictions and the color map describes the time-frequency analysis of HHG. The setup for quantum simulations is the same as Fig. 4.5. 49 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG Figure 4.7: The comparison for semi-classical predictions for excitation occurring (a) at exactly Γ point and (b) in a region within 0.1 a.u. from Γ point. The purple circles describe the semi-classical predictions and the color map describes the time-frequency analysis of HHG. The setup for quantum simulations is the same as Fig. 4.5. k space is important and should be taken into account when studying the dynamics of the solid-phase HHG. 4.1.5 Extension to General Scattering Effects in Solids We have been studying Umklapp scattering based on the one-dimensional quantum simulations and found that Umklapp scattering plays an essential role in the solid-phase HHG. However, Umklapp scattering itself might not be enough in higher-dimensional systems to comprehensively describe the HHG in solids. In solid systems, an electron is not only subjected to Umklapp scattering but also a variety of other scattering mechanisms like scattering with impurity, defect, phonon, or other electrons. Generally speaking, it is very difficult to deal with these complicated scattering events microscopically. Furthermore, while there could only be forwardand backward-scattering in one-dimensional systems, electrons and holes might be scattered into many different angles in twoor three-dimensional systems, which effectively suppresses the recombination of the pairs in the future. Here we propose a very simple modification inspired by the concept of mean-freepath to the generalized three-step model in order to account for the effects of general scattering in solid systems. The idea is quite simple: If an electron-hole pair has traveled a long distance in a crystal, it is in general more likely to be scattered, and its 50 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG chance to recombine in the future would be small. To be more accurate, we made two assumptions. First, it is assumed that the chance of an electron-hole pair undergoing a general scattering event in solids would depend on the distance the pair has traveled. This is reasonable because a large travel distance typically indicates the electron-hole pair has explored a larger volume of the lattice. If the source of scatterings like impurity or defect is evenly distributed in a crystal, the pair with larger volume explored would have a higher chance to encounter a scattering event. Second, we assume the scattered electron-hole pair would have no chance to recombine after the scattering. This supposition should hold as a pair with scattering would have abrupt changes in trajectory so the electron and the hole would have a difficult time meeting each other. Not only that, even if they do eventually meet and emit a photon, this photon emission would be very unlikely to be coherent with photon emission from other electron-hole pairs as the emission timing is overwhelmed by scattering events. Therefore, the photon emission from electron-hole pairs undergoes a general scattering should be of little significance for the final macroscopic HHG. One thing worth mentioning here is that one should not also apply the conclusion of the second assumption here to Umklapp scattering as they are intrinsically different in the sense of coherence. Note that Umklapp scattering discussed in Sec. 3.2 is triggered by the electron-hole pair passing a scattering point in the crystal-momentum space, and the motion of the pair is controlled by the external laser. This means the timing of Umklapp scattering will be controlled by the laser so the photon emissions for these scattered pairs will be coherent with the laser and thereby could be observed in the macroscopic HHG power spectrum. For the actual implementation of the concept into the generalized three-step model, we first define the absolute travel length l(t): l(t) = Zt tex |v(t′)|dt′(4.9) where v(t) is the relative velocity of the electron-hole pair at time t. Also, we will set up a threshold of the absolute travel length lMF P . In the second step of the generalized three-step model, we keep track of the absolute travel length l(t) of an electron-hole pair over time. Once this length becomes larger than the threshold lMF P , the electron-hole pair is considered to encounter a general scattering event and becomes scattered. The recombination and photon emission of this electron-hole pair after the scattering are then neglected (namely its trajectory is terminated). In practice, the values of lMF P will be set to some tens of the lattice constant. We can see that, just like the mean-free path is the travel distance before any interaction, the threshold of the absolute travel length lMFP is the travel distance before any scattering event. This is why our approach here is considered an analogy to the mean-free-path approximation. The cutoff energy dependence on the wavelength of the laser field with lMF P being 3, 5, and 7 nm are described in Fig. 4.8 as red-dashed, green-dotted, and blue-dashedand-dotted curves. Also in the same figure, the result without scattering is shown by the black-solid curve, which is equivalent to the case of lMF P =∞. From Fig. 4.8, one 51 4.1. THE EFFECT OF UMKLAPP SCATTERING ON SOLID HHG 5 7 9 11 1000 2000 3000 4000 Cut-off energy (eV) Wavelength (nm) lMFP=3 nm lMFP=5 nm lMFP=7 nm No scattering Figure 4.8: Wavelength dependence of the cutoff energy of HHG from the generalized three-step model with general scattering effects. The cutoff energy with lMF P equal to 3, 5, and 7 nm are described as red-dashed, green-dotted, and blue-dashed-and-dotted lines. The result without scattering is also depicted as the black-solid line. The wavelength independence can be clearly observed in cases with finite lMFP . This figure is taken from our publication [6]. 52 4.2. THE EFFECT OF A STATIC ELECTRIC FIELD ON SOLID HHG can see that in the long-wavelength region the cutoff energy from the model with general scattering becomes essentially a flat line, suggesting that it is effectively independent of the wavelength of the laser. Within our model here, this wavelength independence could be understood as that electron-hole pairs having long travel enabled by long-wavelength laser are more likely to be scattered during the acceleration process. As recombination from these long-travel electron-hole pairs, which typically carry higher energy due to longer acceleration time, are suppressed by scattering, the highest emission energy gradually saturates with increased laser wavelength [6]. Such wavelength independence had already been observed in previous experimental studies. The reproduction of this feature also suggests the suppression of recombination resulted from general scattering effects in solids plays an important role in wavelength independence of the solid-phase HHG. 4.2 The Effect of a Static Electric Field on Solid HHG 4.2.1 Setup of the Parabolic Two-band System For the the parabolic two-band model introduced in Sec. 2.2. The band gap Egand reduced effective mass Mrof the parabolic band are set to 9 eV and 0.4 electron mass, respectively. These values are chosen specifically such that the resulting system gives clean spectra in time-frequency analysis. As for the laser, the electric field E(t) considered here is in the following form (with ˆzbeing the unit vector in z direction): E(t) = ˆzE0cos4(π(t−τ/2 τ)) cos(ω(t−τ/2)) + ˆzEs(4.10) with the oscillating amplitude E0= 0.4 V/˚ A, photon energy ℏω0= 0.5 eV, and full pulse width T= 60 fs. The static amplitude Esranges from 0 to 0.4 V/˚ A. The electric field E(t) above is valid within the period t∈[0, τ] and is set to zero outside of the period. For numerical simulations of the three-dimensional quantum system, the axis kz, which is the corresponding axis of zaxis in reciprocal space, is discretized with 4096 grid points. The axis kr, which is the radial axis of kz, is also discretized by 4096 grid points. The maximum crystal momentum in kzand krdirection are both set to 4 a.u. In the time propagation, the time step ∆tis set to 0.08 a.u. and a total of 31200 time steps are propagated (total time is around 60.34 fs). As for the semi-classical calculations, the total number of trajectories within the full pulse duration is set to 3000 and the time step ∆tRK is set to 0.01 fs. 4.2.2 The HHG Spectrum under a Static Field We will first examine the differences induced by a static field in a solid-phase HHG power spectrum. In Fig. 4.9, the spectra generated under different strength of the static field are shown. The results for static electric field Esbeing 0, 0.04, 0.08, 0.12, 0.16, and 0.20 V/˚ A are described, respectively, as black, blue, light-blue, pink, red, and brown lines. From this figure, it is observed that, when the strength of the static 53 4.2. THE EFFECT OF A STATIC ELECTRIC FIELD ON SOLID HHG field becomes stronger, the maximum emission energy for the HHG is also increased but the overall intensity with emission energy above band gap is lowered by the static field. Additionally, we perform time-frequency analysis to verify the applicability of Figure 4.9: The power spectra of HHGs under an additional static electric field, with various colors indicating different static field strength. The blue, light-blue, pink, red, and brown lines are results with the static electric field Esbeing 0.04, 0.08, 0.12, 0.16, and 0.20 V/˚ A, respectively. Also, the power spectrum of HHGs without an additional static field is represented as the black line. the semi-classical model for the HHG under an additional static field. In the analysis, the Gabor transformation with a time window of which full width at half maximum is 1 fs is applied to the current obtained from quantum simulations. This result is given in Fig. 4.10, with the quantum result shown as the color map and semi-classical predictions as pink dots. The nice agreement between the results of quantum simulations and semiclassical predictions suggests the generalized three-step model is still applicable when a static field is appended to the system. As a result, we can facilitate the idea of classical trajectories to describe the dynamics of the HHG with a static field. Empowered by the semi-classical model, we could describe the two features observed in the HHG power spectrum, namely higher cutoff energy and lowered overall intensity beyond band gap energy, by some simple physical pictures. The higher maximum emission energy induced by the static field could be understood as the following: In the case with no static field, some high-energy electron-hole pairs are unable to recombine after the oscillatory laser changes its direction because the laser force in this altered direction is not strong enough to bring the pairs back together for recombination. However, in the case when an additional static field presents, if this static force is in the same direction 54 4.2. THE EFFECT OF A STATIC ELECTRIC FIELD ON SOLID HHG Figure 4.10: The time-frequency analysis of HHG (color map) and its semi-classical prediction(pink circles) with the electric field amplitude of the oscillatory laser E0= 0.4 V/˚ A and the electric field amplitude of the static field Es= 0.12 V/˚ A. as the altered oscillatory laser force most of the time, the total force would push the paired electron and hole toward each other. Therefore, these high-energy electron-hole pairs now have higher chances to recombine, emit high-energy photons, and extend the cutoff energy in the HHG power spectrum. As for the lowered overall HHG intensity beyond band gap energy, it could be simply understood by the fact that in general the additional static field is trying to separate electrons and holes away from each other and thereby impedes recombination and lowers the emission intensity. In the extreme case when the strength ratio Es/E0= 1, there cannot be any recombination at all within the semi-classical model because the total electric force exerting on the electron-hole pair is always pointing in one direction, meaning the electron and the hole are always being accelerated away from each other. That is to say, in order to have possible recombination, the oscillatory electric field should overcome the static electric field in some time windows. And it is the electron-hole pairs excited in these time windows that could possibly recombine in the future after the total electric force changes its direction. It is clear that these time windows will gradually diminish as Es/E0increases from 0 and eventually disappear at Es/E0= 1. Consequently, the possibility for recombination also decreases with the increasing static field, yielding a lower HHG intensity. From the two discussions mentioned above, one can see that two mechanisms induced by the static field are competing with each other: Electron-hole recombination of some 55 Chapter 5 Light Absorption of Core Electrons In recent years, the development of free-electron laser (FEL) has created new opportunities for X-ray spectroscopy [44,45]. The generation of high-energy photons in the energy range of the soft X-ray to the hard X-ray in FEL has been realized and demonstrate in various facilities like FLASH, FERMI, SCALA, and LCLS [46,47,48,49]. One of new possibility in X-ray spectroscopy enabled by FEL is harmonic generation using a soft Xray as a seed. In experiments, second harmonic generation from soft X-ray are observed and its intensity dependence on different laser parameters are measured [50,51,52]. The trend in developing harmonic generation with a soft X-ray motivates us to consider the prospect of HHG in solids from an input FEL in the soft X-ray regime. Specifically, we are interested in developing the theoretical framework modeling the HHG using a soft X-ray in solid systems. There are several numerical and theoretical challenges in establishing a practical model for the dynamical process of HHG using a soft X-ray in solids, and one of the major challenges is the appropriate description for light absorption of core electrons. Due to the large photon energy of the X-ray FEL, it is the core electrons deep inside atoms rather than the valence electrons near Fermi level that are excited by the laser. To numerically describe these core electrons which concentrates in a tiny space near a nucleus, a high density of grid points is needed near the nucleus with the real-space grid representation. This leads to very high computational cost for simulations of such processes. Therefore, how efficient a numerical approach is also matters a lot in the choice of theoretical framework. In addition to the numerical challenge, there are also many theoretical difficulties to overcome for appropriately describing the solid-phase HHG using a soft X-ray. For instance, in TDDFT, which is an efficient ab initio model with effective electron-electron interaction [39], the reproduction of light-absorption for coreelectrons is challenging with conventional approximations of the exchange-correlation functionals. In the conventional mean-field treatment, the occupation of core electrons is treated effectively in an averaged sense. Similarly, the core holes created from excitation of core electrons is also treated in the averaged sense. This means the number of core 62 5.1. SATURABLE ABSORPTION IN ALUMINUM AND ITS MECHANISM holes in a cell could be a fractional value, while in real world it is always an integer. Here it should be emphasized that we are talking about the number of core holes in one cell, not the number of core holes per cell calculated from the ratio of the number of core holes to the number of cells in a crystal. In the real world, a fractional value of, say 0.5, for the number of core holes per cell simply means (assuming one core electron per cell) half of cells have one core hole and the rest of the cells has no core hole. Therefore, in one cell the number of core holes is always an integer. However, in the mean-field approximation, a fractional value of 0.5 for the number of core holes per cell would mean every cell has one-half of core holes. This might be of an issue for accurate description for light absorption of core-electron because it has been well-known that the photon absorption in solids depends strongly on the number of the core holes under a high-intensity soft X-ray. The absorption of soft X-ray photons would gradually decreases with a stronger laser so the absorption eventually saturates. This phenomenon is typically know as saturable absorption or light-induced transparency. In order to theoretically describe the solid HHG using a soft X-ray, we need to examine if the absorption of core electrons could be accurately captured in an efficient model. If a model successfully reproduces the saturable absorption of core electrons under soft X-ray, it would thereby serves as a possible candidate for simulating the solid-phase HHG using a soft X-ray. For our study in this chapter, we simulate a solid system of bulk aluminum by employing TDDFT with the adiabatic local density approximation and obtain the absorption coefficient at different intensity of a soft X-ray laser. Then a pulse propagation in an aluminum film in the classical limit is carried out by using the intensity-dependent absorption coefficient to see whether the model could reproduce the saturable absorption. It is found that TDDFT does successfully capture the phenomenon. Also, with a deeper examination on the process, it is also confirmed that the model reproduces the saturable absorption in accordance with the already-established mechanism. Our results here verify that TDDFT could be suitable for the theoretical framework describing solid-phase HHG using a soft-ray, at least for the very first step, namely excitation, of the process. This chapter is composed of the following: In the first section we introduce the saturable absorption and its mechanism behind. Then we discuss the setup of the system and theoretical framework in the second section. Specifically, the microscopic model TDDFT for computing absorption coefficients and the macroscopic model for propagation of a soft X-ray pulse are discussed. In the last section, the result of saturable absorption and correction with some real-world effects are presented. 5.1 Saturable Absorption in Aluminum and its Mechanism Saturable absorption is the behavior of decreased light absorption under a laser of increased intensity. This phenomenon could make a material which is originally opaque to a laser gradually become transparent when the laser intensity becomes stronger, and therefore is sometimes addressed as light-induced transparency. Saturable absorption of a soft X-ray in aluminum has been well-studied and the mechanisms behind the satura63 5.2. THEORETICAL FRAMEWORK tion are also described by a simple model in the previous study [53]. It is proposed that the core electrons which are responsible for absorbing the incoming high-energy photon could be depleted by the high-intensity soft X-ray since the thermal relaxation of excited electrons is not fast enough to supply electrons back to vacancies in core energy levels. When these core electrons are all excited, there are simply no energy receiver for incoming photons, leading to saturable absorption of the light. At the same time, during the excitation of core electrons, the attractive force between core electrons and their parent ion core is also modified, leading to the shift of the core energy levels and rendering absorption of high-energy photon unavailable. The idea of this modified attractive force is that, when core electrons gradually become excited and move away from the parent ion, the electronic shielding formed by the negatively-charged core electrons on the positively-charged parent ion will also become weaker. This means the attractive force from the parent ion acting on the remaining bounded electrons will become larger with the excitation of core electrons. That is, the remaining electrons are now more tightly bounded. Generally speaking, core electrons will experience a stronger increase in attractive force compared to electrons in conduction bands when the shielding is attenuated, because core electrons are more closer to the ion core and thereby more sensitive to the weaken shielding. The energy shift of core levels (2p orbitals) and the conduction band are schematically shown in Fig. 5.1 (a) and (b). As a result, once many core electrons are excited, it would take more energy to move the remaining core electrons to the conduction band at Fermi surface. With the energy shift, the energy of incoming photons of the soft X-ray now is no longer sufficient to excite the morebounded core electrons, leading to saturable absorption. This idea is also illustrated schematically in Fig. 5.1 (c) and (d). In our examination of core absorption, if saturable absorption is indeed reproduced, we should also verify whether the phenomenon is induced by the mechanism described above to ensure the reproduction is not merely a coincidence. 5.2 Theoretical Framework Here we will consider a setup in the experiment [8], namely directing a soft X-ray towards a thin film of aluminum and measuring its transmission. Our theoretical framework would involve a microscopic simulation based on TDDFT [39] and a macroscopic calculation of electronic wave propagation in aluminum based on electrodynamics in the classical limit. We perform microscopic simulations to measure the absorption coefficient under a soft X-ray with different laser intensities. And Then we use this intensitydependent absorption coefficient to calculate attenuation in intensity macroscopically for a soft X-ray pulse propagating in the Al film. Therefore, the soft X-ray pulse penetrating the Al film could be obtained and its transmission through the film can then be calculated. In the procedures mentioned above, we have made several assumptions. First, it is assumed that, when the laser pulse penetrates through the Al film, the change of laser intensity due to absorption over hundreds of lattice sites is tiny. This is equivalent to 64 5.2. THEORETICAL FRAMEWORK Absorption of Photon No Absorption of Photon Bands Before Core Excitations Energy Shift Fermi Energy Bands After Core Excitations (a) (b) (c) (d) Figure 5.1: A schematic view for the mechanism of saturable absorption under a soft X-ray in aluminum. The conduction band is represented by a parabola and the core energy level for 2p orbitals is described by a flat line. Also, an incoming X-ray photon is depicted by the yellow lightning. The energy bands (a) before and (b) after excitation of core electrons are shown in the upper section. In (a) the positive core ion is electronically shielded by core electrons. In (b), the electronic shielding of the core ion is attenuated since some core electrons are excited, and the energy bands are now bounded more tightly, which is indicated by the energy shift of the bands. The interaction between photon and a core electron (c) before and (d) after the bands are shifted is also described in the lower section. (c) Before the energy shift, core electrons can absorb the soft Xray photon and be excited to the Fermi surface. (d) After the energy shift, the photon energy of the soft X-ray is not large enough to make the excitation possible. 65 5.2. THEORETICAL FRAMEWORK cutting the Al film into many even-thinner slices with a thickness of several hundreds of the lattice constant, and assume the soft X-ray is hardly absorbed within each slice. Note that the cut is made in the way such that the soft X-ray has normal incidence on each slice. The assumption allows us to perform the microscopic simulations under an electric field with fixed maximum amplitude. This approximation would be reasonable as long as the thickness of the slice is made small to guarantee little absorption within the slice. Second, within each slice we apply dipole approximation for the laser field to ease the computational cost. This is valid since the wavelength of the soft X-ray used in our study is still much larger than the lattice constant of aluminum. Finally, we treat each slice of the film as a bulk aluminum in the microscopic simulation. The rationale is that, although the slice is quite thin, the thickness is still in the scale of several hundred cells so the light absorption is not sensitive to the two surfaces. 5.2.1 Time-dependent Density Functional Theory For the microscopic simulation, we employ ab initio simulations based on TDDFT, which involves mapping of an interacting many-electron system to many non-interacting single-electron systems [39]. If conventional approximations are applied, the model could be seen as a mean-field approach in the sense that the complicate electron-electron interaction is treated as an average potential depending solely on the density of electrons. In this model, the dynamics of the wavefunction of an electron ψ(r;t) is governed by the time-dependent Kohn-Sham equation: ∂ ∂tϕ(r;t) = H0+HH+Hxc (5.1) where H0,HH, and Hxc are single-electron Hamiltonian, Hatree potential, and exchangecorrelation potential, respectively. The single-electron Hamiltonian contains kinetic energy in the velocity gauge with vector potential A(t) and potential energy from the lattice ions U(r): H0=(P−qA)2 2M+U(5.2) with Pthe momentum operator and Mand qthe electron mass and charge (sign included). The Hartree potential describes the energy from classical Coulomb interaction between the current electron and other electrons: HH=Zr′ q2n(r′) |r−r′|dr′(5.3) with n(r) then density of all electrons. For the exchange-correlation potential Hxc = Hx+Hc, we take the local-density approximation so the exchange part of the potential Hxis taken from the homogeneous electron gas [54]: Hx=−3 4h3n(r) πi1 3(5.4) 66 5.2. THEORETICAL FRAMEWORK and the correlation part Hcis given in term of a parametrization by Perdew and Wang [55]. The vector potential A(t) of the laser pulse used for calculating absorption coefficient admits the form: A(t) = F0 ω0 cos2(π τ(t−τ 2)) cos(ω0t)ˆx(5.5) where F0is the maximum electric field amplitude, τ= 30 fs is the total pulse length, ℏω0= 85.5 eV is the photon energy of the soft X-ray, and ˆxis a unit vector in the positive-xdirection. Outside the time domain 0 ≤t≤τthe vector potential is set to zero. The energy flux, or fluence Φ, for the pulse can be calculated by: Φ = ϵ0cZτ 0hF0cos2(π τ(t−τ 2)) sin(ω0t)i2dt (5.6) with ϵ0the electric permittivity of vacuum and cthe light speed. In this calculation of fluence we use long-pulse approximation (τ≫2π ω0) for the electric field. Note that the photon energy considered here is different from the experiment [8], because the absorption spectrum of bulk aluminum from our theoretical simulation has a different absorption edge so we shift our photon energy such that the same distance from the edge is achieved. See Fig. 5.3. The numerical simulation of the TDDFT is carried out by the program Octopus [56]. In the simulation, electrons of an Al atom in the 1s and 2s orbitals are treated as frozen electrons not participating the photon absorption. The lattice constant for the cubic cell of Al is set to 4.0485 ˚ A. The spacing for grids in real space is set to 0.15 ˚ A, and the number of grid points in the crystal-momentum space is set to 27 in each dimension. For time propagation, the time step is set to 0.02 ˚ A. 5.2.2 Calculation of Transmission The calculation of transmission of soft X-ray through the Al film is carried out by propagating a classical wave in the aluminum film with a fluence-dependent absorption coefficient from the microscopic simulations. We first perform TDDFT simulations under a soft X-ray pulse and measure the excess energy density ρex of bulk aluminum, which is the increment in total energy of electrons per unit volume after the pulse. This quantity could also be interpreted as the absorbed energy per unit area per penetration depth of the advancing pulse. Therefore, by definition, we can calculate the absorption coefficient aby: a=−1 ϕ(0) dϕ(z) dz =1 ϕ(0)ρex (5.7) where ϕ(z) is the energy flux, or fluence, at a penetration depth zof the pulse. Here we assume the pulse is propagating in the positive-zdirection and the Al film starts at z= 0 and ends at z=Lwith L= 53 nm the thickness of the film. 67 5.3. ABSORPTION OF CORE ELECTRONS Figure 5.2: Comparison of the transmission as a function of laser fluence between our theoretical calculation (blue curve) and the experimental data (purple curve) taken from the study [8]. Note that, since only the laser intensity is altered in the pulse and the laser fluence is proportional to the intensity, the axis of fluence can be seen as the axis of intensity. By sampling the absorption coefficients at different intensities (or fluence) of the soft X-ray pulse from microscopic simulations, we can form the absorption coefficients as a function of fluence a(ϕ) by fitting the sampling points with cubic splines. Then the fluence at different penetration depth can be solved by numerical integration of Eq. 5.7. Numerically, we calculate the fluence at the next step ϕ(z+ ∆z) from the fluence at current step ϕ(z) by: ϕ(z+ ∆z) = ϕ(z)−ϕ(0)a(ϕ(z))∆z. (5.8) Finally, the transmission Tis calculated as the ratio of the fluence before and after the Al film, namely T=ϕ(L) ϕ(0) . 5.3 Absorption of Core Electrons 5.3.1 Saturable Absorption The calculated transmission of a soft X-ray at different laser fluence (or intensity) is given in Fig. 5.2 as the blue curve, together with the experimental data (purple curve) for comparison. From this figure, it is clear that our approach not only reproduces the 68 5.3. ABSORPTION OF CORE ELECTRONS Figure 5.3: The absorption spectrum of aluminum near the LII,III absorption edge from theoretical calculation (red curve) and experimental data (blue curve) taken from the previous study [58]. The vector potential of the soft X-ray used in this chapter is also shown as a sharp yellow peak to illustrate its position in the absorption spectrum. The FWHM of the laser is around 0.27 eV. Additionally, we also mark the energy difference of 19 eV between the absorption edge and the laser in the case of our simulation and the experiment. We select a soft X-ray with photon energy of 85.5 eV instead of 92 eV used in the experiment because it has the same distance from the edge in the absorption spectrum. behavior qualitatively but also gives comparable values to the actual experimental data. Most importantly, the transmission of the soft X-ray pulse exhibits clear increment when the laser intensity is increased. This suggests that the Al film becomes more transparent when subjected to a stronger soft X-ray and therefore the saturable absorption is indeed reproduced. To examine the fundamental reason behind the increasing transmission under strong X-ray, we also calculate the absorption spectrum of bulk aluminum. The absorption spectrum can be obtained by sending a kicks (a Dirac delta function in time for the electric field) to the bulk aluminum system and measuring the current density responding to the kick [57]. In Fig. 5.3, the absorption spectrum of bulk aluminum near LII,III absorption edge is shown. The experimental data is also plotted in the same figure for ease of comparison. Despite some differences in the position of peaks, the absorption spectrum from the TDDFT simulation with a kick reasonably capture the overall structure of the experimental spectrum. Note that the photon energy of the soft X-ray is well above the absorption edge for this absorption spectrum from aluminum in the ground state. With 69 5.3. ABSORPTION OF CORE ELECTRONS Figure 5.4: The decrease in energy as a function of fluence for the core level (blue curve) and a conduction band (red curve) compared to their original energy in the ground state. The arrow indicates the fluence around 5 J/cm2where the core level has huge energy shift. Note that at this large energy shift in the core level the energy shift in the conduction band is negligible. This makes the photon energy of the soft X-ray off-resonance and thereby saturable absorption occurs. this kick technique for calculating absorption spectra, we measure the spectrum of the bulk aluminum after the application of the soft X-ray pulse at different pulse intensities. It is found that the general structure of the absorption spectrum remain roughly the same at different laser intensities, but the structure is shifted to region of higher energy. The shift could be made more clear by calculating the energy bands after the application of the soft X-ray pulse. In Fig. 5.4, the change in energy for the core level and the conduction band compared to their ground-state counterparts are described by the blue curve and red curve, respectively. From the figure, it is clear that the core-level energy is much more sensitive to the change of laser fluence compared to the conduction band energy. In the high fluence region (fluence larger than 5 J/cm2), the core level has shifted down in energy by around 20 eV while the conduction band only shifts down by less than 1 eV. This indicates that the energy difference between the core level and the Fermi surface would increase by around 19 eV, or equivalently the LII,III absorption edge shift up in energy by around 19 eV. With this shift, the photon energy of the soft X-ray is no longer above the absorption edge, and therefore the photon can no longer be absorbed. This is the microscopic description for the saturation in absorption and the increase in transmission under stronger soft X-ray. As a quick summary, the behavior of saturable absorption of a soft X-ray in alu70 5.3. ABSORPTION OF CORE ELECTRONS minum is indeed reproduced by the our theoretical framework that combines microscopic TDDFT simulations and macroscopic calculation of classical wave propagation. The success comes from the proper description of the attenuated absorption coefficient under a stronger soft X-ray. Also, by calculation of the energy structure after the pulse, the core energy levels exhibit a large negative shift, leading to an overall shift of the absorption spectrum to a higher energy region and rendering photon absorption unavailable. The mechanism responsible for saturable absorption thus aligns with the description proposed by previous study [53]. This could serve as an evidence for that TDDFT can accurately describe the dynamical process of photon absorption of core electrons in aluminum under a soft X-ray. 5.3.2 Corrections to Transmission from Theoretical Calculation In this final section of the chapter, we would like to briefly consider some real-world effects in experiments to improve the quality of the transmission from our theoretical calculations. For example, in experiment the laser intensity at the focal spot will not be uniform over the tiny shined area. We could assume a simple Gaussian distribution of the laser intensity in the focal spot to estimate this effect. In Fig. 5.5, we have calculated the transmission averaged over the focal spot (red curve), together with the one without focal-spot average (blue curve) and the one from the experiment (purple curve) for comparison. We can see that the transmission as a function of fluence indeed becomes more closer to the experimental data. In addition to the effect of focal-spot average, we could also consider the oxidation of the aluminum film in real experiments. Typically speaking, the surface of the aluminum film will inevitably suffers from oxidation due to the very dilute oxygen remaining in a gas chamber. The oxidized compound, namely Al2O3, has a different absorption coefficient and could lead to some variation of the overall transmission. To assess the effect of the surface oxidation, we add a 10 nm aluminum oxide layer on the front and back side of the 53 nm Al film (so the thickness of the film is totally 73 nm) and the absorption coefficient of the aluminum oxide is set to 3.95 ×105/cm for the soft X-ray. This absorption coefficient is assumed to remain constant under various laser fluence. We show the transmission of such a surface-oxidized Al film (red curve) in Fig. 5.6, together with the one without surface oxidation (blue curve) and the one from the experiment (purple curve). It is clear that the overall transmission is decreased, because the surface oxide layer keeps absorbing the soft X-ray irrespective of the laser intensity. From this result, one could say such surface oxidation does make a big difference so special care should be taken for observing saturable absorption in aluminum. 71 the road. At this rate, I believe the development of HHG would continue to enable the creation of even stronger and sharper illuminating pulses that enlighten the darkness beyond the front line of human knowledge. 78 Appendix A Proof of the Equation In this appendix, we will prove the equation utilized in the derivation of the simple two-band parabolic model discussed in Chapter 2. This equation is (for m=n): ZΩC u∗ mK(t)(r)∂ ∂K(t)unK(t)(r)dr=−ℏ EmK(t)−EnK(t)ZΩC u∗ mK(t)(r)−iℏ∇ MunK(t)(r)dr (A.1) which is actually a special case of the general relation: ∂EnK(t) ∂K(t)δmn =ℏ ΩCZΩC u∗ mK(t)(r)−iℏ∇ MunK(t)(r)dr+δmn K(t)ℏ2 M + [EmK(t)−EnK(t)]1 ΩCZΩC u∗ mK(t)(r)∂ ∂K(t)unK(t)(r)dr(A.2) with m=n. This could be shown true by starting with K(t) derivative of Eq. 2.20 and the followings: ∂ ∂K(t)[EnK(t)unK(t)(r)] = ∂ ∂K(t)[H[K(t)]unK(t)(r)] (A.3) ∂EnK(t) ∂K(t)unK(t)(r) + EnK(t) ∂unK(t)(r) ∂K(t)=ℏ−iℏ∇+ℏK(t) MunK(t)(r) + H[K(t)] ∂unK(t)(r) ∂K(t). (A.4) 80 Then we multiple umK(t)ron both sides and integrate over one cell in real space: ∂EnK(t) ∂K(t) 1 ΩCZΩC u∗ mK(t)(r)unK(t)(r)dr+EnK(t) ΩCZΩC u∗ mK(t)(r)∂unK(t)(r) ∂K(t)dr =ℏ ΩCZΩC u∗ mK(t)(r)−iℏ∇+ℏK(t) MunK(t)(r)dr+1 ΩCZΩC u∗ mK(t)(r)H[K(t)] ∂unK(t)(r) ∂K(t)dr =ℏ ΩCZΩC u∗ mK(t)(r)−iℏ∇ MunK(t)(r)dr+ℏ2K(t) M 1 ΩCZΩC u∗ mK(t)(r)unK(t)(r)dr+ EmK(t) ΩCZΩC u∗ mK(t)(r)∂unK(t)(r) ∂K(t)dr. (A.5) Note that 1 ΩCZΩC u∗ mK(t)(r)unK(t)(r)dr=δmn so we simply have: δmn ∂EnK(t) ∂K(t)+EnK(t) ΩCZΩC u∗ mK(t)(r)∂unK(t)(r) ∂K(t)dr =ℏ ΩCZΩC u∗ mK(t)(r)−iℏ∇ MunK(t)(r)dr+δmn ℏ2K(t) M+EmK(t) ΩCZΩC u∗ mK(t)(r)∂unK(t)(r) ∂K(t)dr. (A.6) By collect similar terms, Eq. A.2 is retained. 81 Bibliography [1] A. Baltuˇska et al. “Attosecond control of electronic processes by intense light fields”. en. In: Nature 421.6923 (Feb. 2003). Number: 6923 Publisher: Nature Publishing Group, pp. 611–615. issn: 1476-4687. doi:10.1038/nature01414.url: https://www.nature.com/articles/nature01414 (visited on 01/08/2021). [2] J. Itatani et al. “Tomographic imaging of molecular orbitals”. en. In: Nature 432.7019 (Dec. 2004). Number: 7019 Publisher: Nature Publishing Group, pp. 867– 871. issn: 1476-4687. doi:10.1038/nature03183.url:https://www.nature. com/articles/nature03183 (visited on 01/08/2021). [3] Eleftherios Goulielmakis et al. “Real-time observation of valence electron motion”. en. In: Nature 466.7307 (Aug. 2010). Number: 7307 Publisher: Nature Publishing Group, pp. 739–743. issn: 1476-4687. doi:10.1038/nature09212.url:https: //www.nature.com/articles/nature09212 (visited on 01/08/2021). [4] G. Vampa et al. “Theoretical Analysis of High-Harmonic Generation in Solids”. In: Phys. Rev. Lett. 113 (7 Aug. 2014), p. 073901. doi:10.1103/PhysRevLett.113. 073901.url:https://link.aps.org/doi/10.1103/PhysRevLett.113.073901. [5] G. Vampa et al. “Semiclassical analysis of high harmonic generation in bulk crystals”. In: Phys. Rev. B 91 (6 Feb. 2015), p. 064302. doi:10.1103/PhysRevB.91. 064302.url:https://link.aps.org/doi/10.1103/PhysRevB.91.064302. [6] Chang-Ming Wang et al. “Role of electron scattering on the high-order harmonic generation from solids”. In: Phys. Rev. Research 2 (3 Aug. 2020), p. 033333. doi: 10.1103/PhysRevResearch.2.033333.url:https://link.aps.org/doi/10. 1103/PhysRevResearch.2.033333. [7] Takuya Ikemachi et al. “Trajectory analysis of high-order-harmonic generation from periodic crystals”. In: Phys. Rev. A 95 (4 Apr. 2017), p. 043416. doi:10. 1103 / PhysRevA . 95 . 043416.url:https : / / link . aps . org / doi / 10 . 1103 / PhysRevA.95.043416. [8] Bob Nagler et al. “Turning solid aluminium transparent by intense soft X-ray photoionization”. In: Nature Physics 5.9 (2009), p. 693. [9] Theodore H Maiman. “Stimulated optical radiation in ruby”. In: nature 187.4736 (1960), pp. 493–494. 83 BIBLIOGRAPHY [10] Theodore H Maiman. “Stimulated optical emission in fluorescent solids. I. Theoretical considerations”. In: Physical Review 123.4 (1961), p. 1145. [11] M Ferray et al. “Multiple-harmonic conversion of 1064 nm radiation in rare gases”. In: Journal of Physics B: Atomic, Molecular and Optical Physics 21.3 (Feb. 1988), pp. L31–L35. doi:10.1088/0953-4075/21/3/001.url:https://doi.org/10. 1088/0953-4075/21/3/001. [12] P. B. Corkum. “Plasma perspective on strong field multiphoton ionization”. In: Physical Review Letters 71.13 (Sept. 1993). Publisher: American Physical Society, pp. 1994–1997. doi:10.1103/PhysRevLett.71.1994.url:https://link.aps. org/doi/10.1103/PhysRevLett.71.1994 (visited on 03/28/2020). [13] M. Lewenstein et al. “Theory of high-harmonic generation by low-frequency laser fields”. en. In: Physical Review A 49.3 (Mar. 1994), pp. 2117–2132. issn: 1050-2947, 1094-1622. doi:10.1103/PhysRevA.49.2117.url:https://link.aps.org/doi/ 10.1103/PhysRevA.49.2117 (visited on 12/21/2019). [14] G. Steinmeyer et al. “Frontiers in Ultrashort Pulse Generation: Pushing the Limits in Linear and Nonlinear Optics”. en. In: Science 286.5444 (Nov. 1999). Publisher: American Association for the Advancement of Science Section: Special Reviews, pp. 1507–1512. issn: 0036-8075, 1095-9203. doi:10.1126/science.286.5444. 1507.url:https://science.sciencemag.org/content/286/5444/1507 (visited on 01/13/2021). [15] Thomas Brabec and Ferenc Krausz. “Intense few-cycle laser fields: Frontiers of nonlinear optics”. In: Rev. Mod. Phys. 72 (2 2000), pp. 545–591. doi:10.1103/ RevModPhys.72.545.url:https://link.aps.org/doi/10.1103/RevModPhys. 72.545. [16] Ferenc Krausz and Misha Ivanov. “Attosecond physics”. In: Rev. Mod. Phys. 81 (1 2009), pp. 163–234. doi:10.1103/RevModPhys.81.163.url:https://link. aps.org/doi/10.1103/RevModPhys.81.163. [17] Thomas Gaumnitz et al. “Streaking of 43-attosecond soft-X-ray pulses generated by a passively CEP-stable mid-infrared driver”. In: Opt. Express 25.22 (2017), pp. 27506–27518. doi:10.1364/OE.25.027506.url:http://www.opticsexpress. org/abstract.cfm?URI=oe-25-22-27506. [18] M. Uiberacker et al. “Attosecond real-time observation of electron tunnelling in atoms”. en. In: Nature 446.7136 (Apr. 2007). Number: 7136 Publisher: Nature Publishing Group, pp. 627–632. issn: 1476-4687. doi:10.1038/nature05648.url: https://www.nature.com/articles/nature05648 (visited on 01/14/2021). [19] Shambhu Ghimire et al. “Observation of high-order harmonic generation in a bulk crystal”. In: Nature physics 7.2 (2011), pp. 138–141. [20] Shambhu Ghimire and David A Reis. “High-harmonic generation from solids”. In: Nature Physics 15.1 (2019), pp. 10–16. 84 BIBLIOGRAPHY [21] T. T. Luu et al. “Extreme ultraviolet high-harmonic spectroscopy of solids”. en. In: Nature 521.7553 (May 2015). Number: 7553 Publisher: Nature Publishing Group, pp. 498–502. issn: 1476-4687. doi:10.1038/nature14456.url:https://www. nature.com/articles/nature14456 (visited on 03/31/2020). [22] G. Vampa and T. Brabec. “Merge of high harmonic generation from gases and solids and its implications for attosecond science”. en. In: Journal of Physics B: Atomic, Molecular and Optical Physics 50.8 (Mar. 2017). Publisher: IOP Publishing, p. 083001. issn: 0953-4075. doi:10.1088/1361-6455/aa528d.url:https: //doi.org/10.1088%2F1361-6455%2Faa528d (visited on 03/28/2020). [23] Mengxi Wu et al. “High-harmonic generation from Bloch electrons in solids”. In: Phys. Rev. A 91 (4 Apr. 2015), p. 043839. doi:10.1103/PhysRevA.91.043839. url:https://link.aps.org/doi/10.1103/PhysRevA.91.043839. [24] Mengxi Wu et al. “Multilevel perspective on high-order harmonic generation in solids”. In: Phys. Rev. A 94 (6 Dec. 2016), p. 063403. doi:10.1103/PhysRevA. 94.063403.url:https://link.aps.org/doi/10.1103/PhysRevA.94.063403. [25] C. R. McDonald et al. “Interband Bloch oscillation mechanism for high-harmonic generation in semiconductor crystals”. In: Physical Review A 92.3 (Sept. 2015). Publisher: American Physical Society, p. 033845. doi:10.1103/PhysRevA.92. 033845.url:https://link. aps.org/doi/ 10.1103/PhysRevA. 92.033845 (visited on 03/31/2020). [26] Georges Ndabashimiye et al. “Solid-state harmonics beyond the atomic limit”. In: Nature 534.7608 (2016), pp. 520–523. [27] Matthias Hohenleutner et al. “Real-time observation of interfering crystal electrons in high-harmonic generation”. In: Nature 523.7562 (2015), pp. 572–575. [28] D. Golde, T. Meier, and S. W. Koch. “High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical interand intraband excitations”. In: Phys. Rev. B 77 (7 2008), p. 075330. doi:10.1103/PhysRevB. 77.075330.url:https://link.aps.org/doi/10.1103/PhysRevB.77.075330. [29] Daniel Golde et al. “Microscopic theory of the extremely nonlinear terahertz response of semiconductors”. In: physica status solidi (b) 248.4 (2011), pp. 863– 866. doi:https : / / doi . org / 10 . 1002 / pssb . 201000840. eprint: https : / / onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.201000840.url:https: //onlinelibrary.wiley.com/doi/abs/10.1002/pssb.201000840. [30] Olaf Schubert et al. “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations”. In: Nature photonics 8.2 (2014), pp. 119–123. [31] G Vampa et al. “Linking high harmonics from gases and solids”. In: Nature 522.7557 (2015), pp. 462–464. 85 BIBLIOGRAPHY [32] Guangjiu Zhao et al. “Attosecond pulse generation by applying a weak static electric field to a few-cycle pulse”. In: New Journal of Physics 13.9 (2011), p. 093035. doi:10.1088/1367-2630/13/9/093035.url:https://doi.org/10.1088/13672630/13/9/093035. [33] Carles Serrat and Jens Biegert. “All-Regions Tunable High Harmonic Enhancement by a Periodic Static Electric Field”. In: Phys. Rev. Lett. 104 (7 2010), p. 073901. doi:10.1103/PhysRevLett.104.073901.url:https://link.aps. org/doi/10.1103/PhysRevLett.104.073901. [34] . “High-order harmonic generation in the presence of a static electric field”. In: Phys. Rev. A 72 (3 2005), p. 033407. doi:10.1103/PhysRevA.72.033407.url: https://link.aps.org/doi/10.1103/PhysRevA.72.033407. [35] Bingbing Wang, Xiaofeng Li, and Panming Fu. “The effects of a static electric field on high-order harmonic generation”. In: Journal of Physics B: Atomic, Molecular and Optical Physics 31.9 (3 1998), pp. 1961–1972. doi:10.1088/0953-4075/31/ 9/012.url:https://doi.org/10.1088/0953-4075/31/9/012. [36] Bogdan Borca et al. “Static-Electric-Field-Induced Polarization Effects in Harmonic Generation”. In: Phys. Rev. Lett. 85 (4 2000), pp. 732–735. doi:10.1103/ PhysRevLett.85.732.url:https://link.aps.org/doi/10.1103/PhysRevLett. 85.732. [37] Shunsuke A. Sato et al. “Role of intraband transitions in photocarrier generation”. In: Phys. Rev. B 98 (3 2018), p. 035202. doi:10.1103/PhysRevB.98.035202.url: https://link.aps.org/doi/10.1103/PhysRevB.98.035202. [38] T. Pfeifer, C. Spielmann, and G. Gerber. “Femtosecond x-ray science”. en. In: Reports on Progress in Physics 69.2 (3 Jan. 2006). Publisher: IOP Publishing, pp. 443–505. issn: 0034-4885. doi:10.1088/0034-4885/69/2/R04.url:https: //doi.org/10.1088%2F0034-4885%2F69%2F2%2Fr04 (visited on 03/28/2020). [39] Erich Runge and E. K. U. Gross. “Density-Functional Theory for Time-Dependent Systems”. In: Phys. Rev. Lett. 52 (12 1984), pp. 997–1000. doi:10.1103/PhysRevLett. 52.997.url:https://link.aps.org/doi/10.1103/PhysRevLett.52.997. [40] Nicolas Tancogne-Dejean et al. “Impact of the Electronic Band Structure in HighHarmonic Generation Spectra of Solids”. In: Phys. Rev. Lett. 118 (8 2017), p. 087403. doi:10.1103/PhysRevLett.118.087403.url:https://link.aps.org/doi/10. 1103/PhysRevLett.118.087403. [41] J. B. Krieger and G. J. Iafrate. “Time evolution of Bloch electrons in a homogeneous electric field”. In: Physical Review B 33.8 (3 Apr. 1986), pp. 5494–5500. doi:10.1103/PhysRevB.33.5494.url:https://link.aps.org/doi/10.1103/ PhysRevB.33.5494 (visited on 12/21/2019). [42] E.O. Kane. “Zener tunneling in semiconductors”. In: Journal of Physics and Chemistry of Solids 12.2 (3 1960), pp. 181–188. issn: 0022-3697. doi:https://doi. org/10.1016/0022-3697(60)90035-4.url:https://www.sciencedirect.com/ science/article/pii/0022369760900354. 86 BIBLIOGRAPHY [44] Primoz Ribic and G. Margaritondo. “Status and prospects of X-ray free-electron lasers (X-FELs): a simple presentation”. In: Journal of Physics D-applied Physics - J PHYS-D-APPL PHYS 45 (3 May 2012), p. 033407. doi:10.1088/00223727/45/21/213001.url:https://link.aps.org/doi/10.1103/PhysRevA.72. 033407. [45] Iwao Matsuda and Yuya Kubota. “Recent Progress in Spectroscopies Using Soft X-ray Free-electron Lasers”. In: Chemistry Letters 50.7 (3 July 2021). Publisher: The Chemical Society of Japan, pp. 1336–1344. issn: 0366-7022. doi:10.1246/ cl.200881.url:https://www.journal.csj.jp/doi/full/10.1246/cl.200881 (visited on 07/05/2021). [46] W. Ackermann et al. “Operation of a free-electron laser from the extreme ultraviolet to the water window”. en. In: Nature Photonics 1.6 (3 June 2007). Bandiera abtest: a Cg type: Nature Research Journals Number: 6 Primary atype: Research Publisher: Nature Publishing Group, pp. 336–342. issn: 1749-4893. doi: 10 . 1038 / nphoton . 2007 . 76.url:https : / / www . nature . com / articles / nphoton.2007.76 (visited on 07/05/2021). [47] E. Allaria et al. “Tunability experiments at the FERMI@Elettra free-electron laser”. en. In: New Journal of Physics 14.11 (3 Nov. 2012). Publisher: IOP Publishing, p. 113009. issn: 1367-2630. doi:10.1088/1367-2630/14/11/113009.url: https://doi.org/10.1088/1367-2630/14/11/113009 (visited on 07/05/2021). [48] Tetsuya Ishikawa et al. “A compact X-ray free-electron laser emitting in the sub- ˚angstr¨om region”. en. In: Nature Photonics 6.8 (3 Aug. 2012). Bandiera abtest: a Cg type: Nature Research Journals Number: 8 Primary atype: Research Publisher: Nature Publishing Group Subject term: Free-electron lasers;X-rays Subject term id: free-electron-lasers;x-rays, pp. 540–544. issn: 1749-4893. doi:10 . 1038/nphoton.2012.141.url:https://www.nature.com/articles/nphoton. 2012.141 (visited on 07/05/2021). [49] P. Emma et al. “First lasing and operation of an˚angstrom-wavelength free-electron laser”. en. In: Nature Photonics 4.9 (3 Sept. 2010). Bandiera abtest: a Cg type: Nature Research Journals Number: 9 Primary atype: Research Publisher: Nature Publishing Group Subject term: Free-electron lasers;X-rays Subject term id: freeelectron-lasers;x-rays, pp. 641–647. issn: 1749-4893. doi:10.1038/nphoton.2010. 176.url:https://www.nature.com/articles/nphoton.2010.176 (visited on 07/05/2021). [50] Sharon Shwartz et al. “X-Ray Second Harmonic Generation”. In: Physical review letters 112 (3 Apr. 2014), p. 163901. doi:10.1103/PhysRevLett.112.163901. url:https://link.aps.org/doi/10.1103/PhysRevA.72.033407. [51] R. K. Lam et al. “Soft X-Ray Second Harmonic Generation as an Interfacial Probe”. In: Physical Review Letters 120.2 (3 Jan. 2018). Publisher: American Physical Society, p. 023901. doi:10 . 1103 / PhysRevLett . 120 . 023901.url: 87