High-order harmonic generation from two-dimensional materials subjected to intense laser fields

doi:10.21203/rs.3.rs-5014226/v1

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High-order harmonic generation from two-dimensional materials subjected to intense laser fields

https://doi.org/10.21203/rs.3.rs-5014226/v1

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Based on the real-time time-dependent density functional theory, we theoretically investigate the influence of bandgap on the high-order harmonic generation (HHG) from monolayer hexagonal two-dimensional (2D) solids: Gallium Phosphide (GaP), Graphene, Borophene (graphene-like), and Boron nitride (h-BN) under a few-cycle linearly- and/or single circularly-polarized laser pulse. Our results show that interband currents are prominently larger in the zigzag (ZZ) direction in comparison with the armchair (AC) direction, when the laser field is polarized along the ZZ-direction. Accordingly, the high-order harmonics can be produced more efficiently along the ZZ-direction than that of the AC-direction. We exhibit that single-layer 2D materials can generate bulk-like high-order harmonics when they are driven by an in-plane polarized laser field, and atomic-like harmonics when driven by an out-of-plane polarized laser field. Our findings indicate that due to the difference in the effective mass of carriers along AC- and ZZ-directions, the high-order harmonics spectra are different in both directions. In addition, the results illustrate that the dependence of HHG intensity changes according to the polarization of the laser electric field. The bandgap significantly affects the HHG, most importantly through ultrafast modification of the interband polarization of the system. Finally, based on the present study, borophene and GaP have outstanding potential for future utilization in extreme-ultraviolet, efficient table-top HHG sources, and as an ultrafast optical tool to provide possibilities for imaging solid structures.

High harmonic generation

monolayer 2D material

laser-matter interaction

DFT

Over the last several decades, with the advent and development of intense femtosecond lasers, the HHG from gaseous targets has become one of the most important research areas in ultrafast optics [1–6]. In this approach, via the nonlinear interaction of an intense mid-infrared (MIR) femtosecond laser with a gaseous target and/or a solid-state system can generate high-order harmonics. The HHG is a main approach to obtain the table-top level coherent radiation source in the extreme ultraviolet (XUV) and soft x-ray region. In recent years, much attention has been attracted to the HHG from condensed-matter systems, because the high density of a solid-state material can give higher conversion efficiency relative to a gas. Therefore, solid-state materials may provide direct access to compact and brighter XUV light sources [7]. The mechanism of HHG from condensed phase systems originates from nonlinear intraband current, and interband polarization, which can be illustrated by the semiclassical three-step model: (i) an electron-hole pair is generated due to the excitation of an electron from the valence bands to the conduction bands under an intense MIR laser field. (ii) the electrons (holes) accelerate on the conduction (valence) bands, which lead to an intraband current. (iii) the electrons recombine with holes in the valence band and a time-dependent interband current is generated. However, the coupling mechanism between interband and intraband processes has not yet been fully understood for the optical response of solid state materials. However, it is well-known that the sum of inter- and intraband currents contribute to the total HHG [7].

Among various solid-state materials, the atomic thickness of 2D materials makes their electronic and optical properties very sensitive to external electric fields. The HHG from 2D materials displays distinct electronic properties relative to the bulk solids. For example, it has been demonstrated that the HHG from isolated single-layer MoS₂ is more efficient than those generated from bulk [8]. Since the HHG from the bulk ZnO was first observed in 2010 [9], the HHG from 2D condensed matters such as graphene [10], silicene [11], black phosphorous [12], transition metal dichalcogenides [13–14], and h-BN [15–16] have been performed experimentally and theoretically. Here, we attempt to study the high harmonics emission from four kinds of 2D materials: Gallium Phosphide (GaP), Graphene, Borophene (Boron sheet), and Boron nitride (h-BN) which all of them arranged in hexagonal honeycomb lattices. Compared to graphene, borophene exhibits unique properties in both fundamental studies and various applications. For example, graphene due to the zero-bandgap, many of its applications are limited, such as photodynamic therapy, bio-imaging, logic circuits, and so on [17 and references therein]. Black phosphorous (BP) is a promising 2D material that overcomes this issue by filling the bandgap between graphene (0 eV) and transition metal dichalcogenides (1.5–2.5 eV), offering broadband saturable absorption from the visible to mid-infrared. Despite the benefits, BP has limited application because it easily degrades into phosphoric acid when it comes in contact with light, oxygen, or water. Therefore, it seems that borophene can be an excellent candidate and an appropriate alternative to graphene, because it can exhibit higher electrical conductivity and strength than graphene [17].

The ellipticity dependence of the HHG from 2D materials such as graphene has been observed experimentally and discussed theoretically. Yoshikawa et al. [18] observed that HHG signal from graphene is enhanced under ellipticity polarized excitation, in contrast, Taucer et al. [19] reported that the ellipticity dependence of the high-order harmonics from graphene is similar to gaseous systems. Saito and his colleagues [20] experimentally demonstrated selection rules for the HHG from hexagonal gallium selenide crystal irradiated by single-circularly and bicircularly-polarized MIR pulses. Ghimire et al. [9] found experimentally that the emitted harmonics from bulk ZnO are less sensitive to the ellipticity of the driving laser field in comparison with HHG from the gaseous medium. They indicated that interband and intraband mechanisms are strongly different by the ellipticity of laser field. Dong et al. [21] revealed theoretically that the harmonic ellipticity depends on the harmonic order and the orientation angle between the graphene symmetric axis and laser polarization direction. As a result, the HHG from solid-state systems by the ellipticity polarized laser field is dependent strongly on solid species. However, compared to gaseous media, there seems no consensus yet that how the ellipticity influences the HHG from the condensed-matter systems. So far, to the best of our knowledge, the important extreme nonlinear behavior and ultrafast dynamics of atomically monolayer borophene and its comparison with other members of 2D materials family, such as GaP, graphene, and insulating h-BN under strong fields have not yet been explored.

In the present work, we focus on the HHG from four kinds of monolayer 2D materials with different bandgap energies (borophene, GaP, graphene, and h-BN) using the first-principal calculations by numerically solving Kohn-Sham equations in the framework of time-dependent density functional theory (TD-DFT). Because of the difference in bandgap, nonlinear behavior and electronic properties of 2D materials can be completely different. Thus, we thoroughly explore the non-perturbative HHG from single-layer 2D materials driven by linearly- and single circularly-polarized MIR laser fields, respectively. Then, the HHG along the armchair (AC) and zigzag (ZZ) directions from the monolayer 2D materials under linearly-polarized laser pulse are systematically investigated. In the following, the influence of in-plane and out-of-plane directions driven by linearly-polarized laser field on the HHG is illustrated. Finally, we explore the HHG from the single-layer 2D materials in the AC- and ZZ-directions under a single circularly-polarized laser pulse and then compare with linear polarization.

The paper is arranged as follows: In Sec. 2, it outlines the computational approach when the HHG from 2D materials under intense laser fields are simulated. In Sec. 3, we report the numerical results of condensed materials harmonic spectra and carry out a detail analysis and discussion. Our main conclusions are summarized in Sec. 4. Atomic units are used in this paper unless otherwise specified.

Within the framework of DFT and TD-DFT, the time-independent and the time-dependent Kohn-Sham equations are numerically solved, respectively [22]. The single-layer 2D materials is exposed to a linearly- and a single circularly-polarized MIR laser field, which are polarized along the in-plane and out-of-plane configurations. The time-dependent Kohn-Sham equation is expressed as

$$\:i\text{ћ}\frac{\partial\:{\psi\:}_{n,\varvec{k}}(\varvec{r},t)}{\partial\:t}={\widehat{H}}_{KS}(\varvec{r},t){\psi\:}_{n,\varvec{k}}\left(\varvec{r},t\right)$$

In the above equation, $\:{\psi\:}_{n,\varvec{k}}$ is a Bloch state, n is a band index, and k is a point in the BZ. The time-dependent Kohn-Sham potential is determined in the following form

$$\:{\widehat{H}}_{KS}(\varvec{r},t)=\frac{1}{2m}{\left[-i{\text{ћ}\nabla\:}^{2}+\frac{e}{C}\varvec{A}\left(t\right)\right]}^{2}+\:{V}_{ion}\left(\varvec{r},t\right)+{V}_{H}\left(\varvec{r},t\right)+{V}_{xc}\left(\varvec{r},t\right)$$

The first term includes the kinetic energy operator and the laser-matter interaction, which is described in the velocity gauge, with m being the effective mass of electron, C denotes the speed of light, and $\:e$ is the elementary charge. $\:{V}_{ion}\left(\varvec{r},t\right)$ represents the ionic potential, $\:{V}_{H}\left(\varvec{r},t\right)$ represents the Hartree potential, $\:{V}_{xc}\left(\varvec{r},t\right)$ is the exchange-correlation potential. $\:{n(\varvec{r},t)=\sum\:_{i}\left|{\psi\:}_{n,\varvec{k}}\left(\varvec{r},t\right)\right|}^{2}$ denotes the time-dependent electron density.

$\:\varvec{A}\left(t\right)$ is the laser vector potential acting on the electrons and is written in the following form:

$${\mathbf{A}}(t)=\frac{{\sqrt {{I_0}} C}}{\omega }f(t)\left[ {\frac{1}{{\sqrt {1+{\varepsilon ^2}} }}cos({\omega _1}t)\widehat {{{{\mathbf{e}}_x}}}+\frac{\varepsilon }{{\sqrt {1+{\varepsilon ^2}} }}sin({\omega _\varvec{2}}t)\widehat {{{{\mathbf{e}}_y}}}} \right]$$

Where ${I_0}$ is the peak intensity of pump laser, ${\omega _1}$ and ${\omega _2}=2{\omega _1}$ are the laser frequencies, $f(t)$ is the laser pulse envelope,$\varepsilon$ represents the ellipticity parameter, ${e_x}$ and ${e_y}$ denote two mutually perpendicular real unit vectors. For the linearly-polarized laser, ellipticity parameter is set to be zero and for the circularly-polarized laser, it is set to be unit. The laser pulse envelope is a sin-square profile and the carrier-envelope phase is taken to be zero (φ = 0). The electric field E(t) relates to the A(t) by ${\mathbf{E}}(t)= - \frac{1}{c}\frac{\partial }{{\partial t}}{\mathbf{A}}(t)$.

The high-order harmonic spectrum is obtained from the time-dependent electronic current density through a Fourier transform [23–27]

$$\operatorname{HHG} (\omega )={\left| {\operatorname{FT} \left[ {\frac{\partial }{{\partial t}}\int\limits_{\Omega } {{\mathbf{J}}({\mathbf{r}},t){d^3}r} } \right]} \right|^2}$$

Here $\Omega$ is the unit cell volume. The time-dependent electronic current density ${\mathbf{J}}({\mathbf{r}},t)$ can be obtained as [28]

$${\mathbf{J}}({\mathbf{r}},t)= - e\sum\limits_{i} {\operatorname{Re} \left[ {\Psi _{i}^{*}({\mathbf{r}},t)\left( { - i\hbar \nabla +\frac{e}{c}{\mathbf{A}}(t)} \right){\Psi _i}({\mathbf{r}},t)} \right]}$$

The number of excited electrons ${\operatorname{N} _{exc}}$ per unit cell is defined by the projections of the time-evolved wave functions (${\psi _{n,{\mathbf{k}}}}(t)$) on the basis of the ground-state wave functions (${\psi _{m,{\mathbf{k}}}}(0)$), which can be written as [29]

$${\operatorname{N} _{exc}}={\operatorname{N} _e} - {\int\limits_{{BZ}} {\sum\limits_{{n,m}}^{{occ}} {{f_{n,{\mathbf{k}}}}\left| {\left\langle {{\psi _{n,{\mathbf{k}}}}(t)|{\psi _{m,{\mathbf{k}}}}(0)} \right\rangle } \right|} } ^2}\operatorname{d} {\mathbf{k}}$$

in which ${\operatorname{N} _e}$ is the total number of electrons in the system and ${f_{n,k}}$ is the occupation of the Kohn-Sham orbitals on the n^th band at point k. All calculations are performed using the real-space grid-based code, Octopus, which has been successfully developed to describe strong-field phenomena in solids [30, 31].

The real-space cell is sampled along the three directions by a uniform grid spacing of 0.4 a.u. and 31×31 Monkhorst-Pack k-point grid is used to sample the 2D Brilluion zone (BZ). During propagation we added a complex absorbing potential (CAP) with a $\:{sin}^{2}\left(z\right)$ shape of width 5 a.u. on the edges of the computational parallelepiped box and CAP height η=-1 a.u. to avoid spurious reflection of electrons at the boundaries [32]. The time-dependent Kohn-Sham equations are solved using the approximate enforced time-reversal symmetry (AETRS) scheme. In the calculations, the self-interaction corrected local-density approximation (SIC-LDA) which introduced by Perdew and Zunger for the electronic exchange-correlation [32], and Hartwigsen-Geodecker-Hutter (HGH) pseudopotentials for core-electron potentials are employed [33]. Atomic units are used in this paper unless otherwise specified.

Simulations are performed for generating of high-order harmonics from the prototype 2D materials. E₀ in atomic units is defined in terms of the peak intensity of the laser (I₀) and is given ${E_0}=\sqrt {{{{I_0}} \mathord{\left/ {\vphantom {{{I_0}} {3.51 \times {{10}^{16}}}}} \right. \kern-0pt} {3.51 \times {{10}^{16}}}}}$, where the peak intensity of the pump laser is 3 TW cm^− 2. All calculations are performed at wavelengths of ${\lambda _1}=1600$nm (${\omega _1}=0.028$ a.u.) and ${\lambda _2}=800$nm (${\omega _1}=0.056$ a.u.), respectively. T is the pulse duration at full width half maximum (FWHM), which is set to be five cycles in this work. The time-dependent Kohn-Sham equations are numerically solved in a real-space grid with spacing of $\Delta {\mathbf{r}}=0.4$a.u.

3.1. The case of linearly-polarized pulse along in-plane direction

We first study the effect linear polarization of interacting laser field on the radiation of high-order harmonics from a single-layer borophene (metallic with near-zero gap) and their comparison with emitted harmonics from representative kinds of monolayer 2D materials, i.e., graphene (semimetallic with zero gap), GaP (semiconductor with 2.26 eV gap), and h-BN (insulator with 4.6 eV gap). It is assumed that the driving laser field is linearly polarized along the x-axis direction. Figure 1(a)-(b) displays the geometrical structure of a hexagonal honeycomb single-layer 2D lattice in real-space and schematic diagram of laser-matter interaction along the in-plane and out-of-plane directions. Different from graphene and borophene, the two atoms in the primitive cell of h-BN and GaP are not the same. During the calculations, the nuclei were fixed, so that the energy transfer from electrons to ions is not considered.

So far, many researchers have investigated the anisotropic properties in 2D condensed matter systems such as electrical conductance, thermal conductance, and linear/nonlinear optical properties etc, however, anisotropic extreme nonlinear optical properties under strong-field excitation are less studied. Figure 2 shows the total electronic currents in the four representative kinds of 2D materials along either the zigzag (ZZ, the x-axis) or armchair (AC, the y-axis), respectively. It exhibits apparent anisotropic properties on the trailing edge of the total electronic currents. The laser field is polarized linearly along the ZZ-direction of the crystals, where $\overline {{\Gamma X}}$ and $\overline {{\Gamma Y}}$ in the BZ, corresponds to the ZZ and AC-directions in the real-space, respectively. For the graphene case, one sees that the electronic current is only generated in the ZZ direction (parallel to the driving laser field). Nonetheless, for the borophene, GaP, and h-BN the electronic currents are generated in both the ZZ- and AC-directions. This discrepancy originates from the two reasons. The first, gapped materials have non-zero Berry curvature due to inversion symmetry breaking. To explain the influence of Berry curvature on similar materials, a simple model was proposed by Liu and coworkers [35]. According to this model, the velocity of an electron in the n^th energy band is given by

$$\mathop {\mathbf{r}}\limits^{{\mathbf{.}}} =\frac{1}{\hbar }\frac{{\partial \varepsilon ({\mathbf{k}})}}{{\partial {\mathbf{k}}}} - \mathop {\mathbf{k}}\limits^{{\mathbf{.}}} \times \Omega ({\mathbf{k}})$$

The first term on the right originates from the band dispersion and the second term stems from the Berry curvature ${\Omega _n}$. Based on this model, Bloch oscillation of electrons in the conduction band leads to a nonlinear current in the crystals parallel to the laser field, while the Berry curvature acts like an out-of-plane magnetic field leading to an anomalous in-plane current perpendicular to the driving field. The second reason is that the energy eigenvalue of the gapped materials can be generally described by

$$\varepsilon ({\mathbf{k}})= \pm \sqrt {{{\left( {{\raise0.7ex\hbox{${{\Delta _g}}$} \!\mathord{\left/ {\vphantom {{{\Delta _g}} 2}}\right.\kern-0pt}\!\lower0.7ex\hbox{$2$}}} \right)}^2}+t_{0}^{2}|f({\mathbf{k}}){|^2}}$$

Here ${\Delta _g}$is the minimum bandgap energy at k-points in the Brillouin zone. Adding a small energy gap essentially breaks the inversion symmetry of the lattice. Graphene is a gapless matter and its lattice has inversion symmetry. Thus, the electronic current disappears in the $\overline {{\Gamma Y}}$ or AC polarization direction (y-axis), since the laser is polarized linearly along the $\overline {{\Gamma X}}$ or ZZ-direction (x-axis).

The high-order harmonic spectra from four kinds of monolayer 2D materials and the number of excited electrons during the laser pulse corresponding to these harmonics are shown in Fig. 3. One can observe that under the same pump condition, the overall HHG efficiency from h-BN is lower than other 2D materials. In contrast, the HHG from borophene is comparable to GaP. On the other hand, the number of excited electrons to conduction bands under a linearly-polarized laser field in h-BN is the lowest value, and the population of conduction bands for both GaP and borophene are the highest and also are comparable to each other. As a consequence, the population of conduction bands decreases drastically as the band gap increases. Taking this into account, the h-BN is an insulator with a width bandgap; thus the excitation rate of electrons to its conduction bands is low. For this reason, the overall HHG efficiency from h-BN is lower than other materials. Furthermore, the number of excited electrons in graphene is smaller than that of GaP and borophene, in spite of that fact graphene has zero bandgap. This can be explained by the density of states (DOS) of the materials. Graphene and h-BN have a vanishing DOS at the Fermi level, while GaP and borophene have a flat band-dispersion near the Fermi level that can lead to a large DOS. Therefore, more electrons from valence bands can be excited to conduction bands in GaP and borophene. Besides number of electrons, the shape of band-dispersion is different in these four kinds of monolayer 2D materials. So, intraband contribution in overall current can be significantly influenced by the shape of band-dispersion. As can be seen from harmonic spectra (Fig. 3), the bandgap width does not affect the shape of HHG spectrum significantly but only weakens slightly the yield of high-order harmonics.

In this subsection, we investigate how the direction of laser-matter interaction affects the HHG from the monolayer 2D solid state systems driven by a linearly-polarized laser field along AC ($\overline {{\Gamma Y}}$) direction and its comparison with ZZ ($\overline {{\Gamma X}}$) direction. Figure 4 illustrates comparison the HHG from the monolayer 2D materials with driving laser field polarized along the AC ($\overline {{\Gamma Y}}$) and the ZZ ($\overline {{\Gamma X}}$) directions. As can be seen from this figure, the yield of high-order harmonics is different in both directions. This is because of the fact that the effective mass of carriers is different along the AC- and ZZ-directions. For instance, in h-BN, the effective mass of electrons along the armchair is 0.83$\:{m}_{e}$ and along the zigzag is 1.15$\:{m}_{e}$, respectively [36]. On the other hand, the mobility of carriers ($\:\mu\:$) is inversely proportional to the effective mass of carriers ($\:\mu\:\sim\frac{1}{{m}^{*}}$). Therefore, it is expected that the mobility of carriers in the armchair direction is larger than the zigzag direction. The effective mass of carriers is related to the dispersion of valence and conduction bands ($\:\frac{{\partial\:}^{2}E}{\partial\:{\varvec{k}}^{2}}$) as follows: $\:\frac{1}{{m}^{*}}=\frac{{\partial\:}^{2}E}{\partial\:{\varvec{k}}^{2}}$, in which E is the band energy, $\hbar$ is the reduced Planck constant, and k is the wave vector. The flat dispersion of the valence and conduction bands results in a smaller coefficient of $\:\frac{{\partial\:}^{2}E}{\partial\:{\varvec{k}}^{2}}$ that leads to a heavier effective mass of carriers. On the other hand, the parabolic dispersion of edge conduces a large coefficient of $\:\frac{{\partial\:}^{2}E}{\partial\:{\varvec{k}}^{2}}$ as a result, carriers with lighter effective mass can be extrapolated. Accordingly, difference in the high-order harmonics spectra between AC- and ZZ-directions is attributed to the anisotropic band-dispersion along AC- and ZZ-directions.

3.2. The Case of a linearly-polarized pulse along the out-of-plane direction

In the following, the total currents are calculated for the out-of-plane polarized laser field (perpendicular to the monolayer’s plane), as demonstrated in Fig. 5. By comparing total currents along the z-axis (out-of-plane case) with x- and y-axis (AC- and ZZ-directions or in-plane case), one finds that the current along the z-axis is dominant to the other axes. In this case, the currents produced along the x- and y-axes are so weak that can be taken as a perturbation. In contrast, for the case of in-plane polarization (Fig. 2), the generated currents are dominant along the x- and y-axes (ZZ- and AC-directions).

In Fig. 6, we exhibit the HHG spectra emitted from representative kinds of single-layer 2D materials for the case of an out-of-plane polarized driving laser field (direction of z-axis) and corresponding the number of excited electrons. As seen from this figure, the number of excited electrons in borophene is smaller than that of GaP, while its harmonics yield is higher than that of GaP. This may be attributed to the fact that fraction of excited electrons do not return to the ground state and remain in the conduction bands and occupy broader areas of BZ, i.e., larger momentum space. As a consequence, these excited electrons generate an intraband current, which contribute to the HHG collectively with an interband current. By comparing Fig. 3 and Fig. 6, it can be seen that the number of excited electrons in the in-plane direction is much greater than that of the out-of-plane direction. When more electrons to be excited to the conduction bands during the interaction with the external electric field, larger electronic currents are generated. Thus, the yield of harmonics is enhanced. As Fig. 3 displays the efficiency of high-order harmonics for the in-plane direction is about three orders larger than those generated by the out-of-plane ones (Fig. 6), because the electronic currents are prominently larger for the case of in-plane polarization than out-of-plane. We conclude the monolayer 2D materials can generate bulk-like high-order harmonics when they are driven by an in-plane polarized laser field, and atomic-like harmonics when driven by an out-of-plane polarized laser field.

3.3. The Case of single circularly-polarized laser pulse

It is well known that when an atomic or molecular target is irradiated by a single circularly-polarized pulse, the electrons can be transversely accelerated and miss the parent ions. Accordingly, the probability of direct recollision is significantly reduced. In contrast, in the solid-state systems, especially for 2D materials due to the multi-center periodic potential wells and delocalization of the Bloch wave packets, an electron ionized from one atomic site in the periodic potentials may recombine with any other site [37]. The HHG from solid-state materials could be produced under a single-circularly-polarized laser pulse so that the plane of laser polarization is the same with the plane of 2D solids [2]. In this subsection, we focus on the generating of high-order harmonics from four kinds of monolayer 2D materials driven by the external single circularly laser field. We keep the incident laser intensity the same as linear polarization. Comparison of the HHG spectra from representative kinds of monolayer 2D materials under linearly- and single circularly-polarized laser field along the ZZ-direction are depicted in Fig. 7. As can be seen from this figure, for monolayer borophene crystal, in the low-energy region (< 15 harmonic orders) of harmonic spectrum, the efficiency of several harmonic orders under linear polarization is higher than those generated under circular polarization; nonetheless, in the high-energy region of the harmonic spectrum, the harmonics generated under circular polarization have higher yield than those generated by linear polarization. For monolayer GaP crystal under linear polarization, except one harmonic in the low-energy region of the harmonic spectrum, all harmonics are generated with a higher yield than those generated in the case of circular polarization. For a single-layer graphene lattice under linear polarization, not only is the harmonic cutoff frequency extended remarkably but also the efficiency of all harmonics is higher than those generated by a circularly polarized pulse. Finally, for a single-layer h-BN lattice under linear polarization, similar to GaP, except some harmonics in the low-energy region of the harmonic spectrum, all harmonics have higher yield compared to harmonics driven by single circularly-polarized pulse. In the following, a comparison of the electronic current densities driven by single circularly-polarized MIR laser field along the AC- and ZZ-directions is illustrated in Fig. 8. The laser fluence, frequencies, and duration are the same as the case of linear field in Fig. 2. In this investigation, 2D materials are located in the X-Y plane as well as the polarization plane of circular laser field is in the X-Y plane. So, the laser field interacts with the 2D materials in both the AC- and ZZ- directions. As seen from these figures, there is a much stronger anisotropic band-dispersion than that linear polarization on the trailing edge of electronic currents in both the AC- and ZZ-directions. It is expected that the electronic currents with relatively equal intensities are generated in both the AC- and ZZ-directions. Accordingly, as shown in Fig. 8, the electronic currents are comparable in both directions for all 2D materials, although the electronic current densities in the borophene and GaP are stronger than other 2D materials. For this reason, the efficiency of high-order harmonics under circular excitation is comparable in both directions (not shown here). Figure 9 demonstrates the HHG spectra from 2D materials under circular polarization and corresponding the number of excited electrons. Under the same pump conditions, it can be noticed that, for single-layer GaP lattice, the efficiency of harmonic orders is higher than other 2D materials; nonetheless, in the high-energy region (> 15 harmonic orders) of the harmonic spectrum, the yield of harmonics generated from borophene is higher than others. In general, when the more fractions of electrons excited from valence bands to conduction bands, a stronger electronic current is generated which will lead to an increase in the yield of harmonics. These findings are in accordance with the results presented in Fig. 8. We conclude the HHG can be efficiently generated in single circularly polarized driving laser field with the polarization plane in the same plane of 2D materials. Our findings are in agreement with the previous experimental results [20, 38].

In conclusion, the real-time real-space TD-DFT methodology is used to obtain the HHG spectra from 2D materials, when these materials are irradiated by single circularly- and/or linearly-polarized laser field. By solving three-dimensional TD-KS equations developed within the framework of TD-DFT, a reliable calculation of dynamics of excited electrons, total currents and high-order harmonics spectra is presented. Using the linearly-polarized laser field along the direction of x-axis, we obtained more efficient high harmonics from 2D materials in the parallel case (ZZ-direction), compared to those generated along the AC-direction. Furthermore, contrary to the absence of HHG from gaseous media, our findings show that the HHG from the interaction between the monolayer 2D materials and single circularly polarized laser field can be produced. Our ab initio simulations reveal that the kind of 2D material and its bandgap have a significant influence on the efficiency of solid-state HHG. Among representative kinds of monolayer 2D condensed matter, the borophene and GaP generate more efficient the HHG and also their results are comparable. Thus, they are outstanding candidates for future utilization in extreme-ultraviolet, efficient table-top HHG sources, and as an ultrafast optical tool to provide possibilities for imaging solid structures.

The authors declare no conflicts of interest.

Funding

The author declares that no fund, grant, or other support was received during the preparation of this manuscript.

Author Contribution

A. M. Koushki wrote the main manuscript text.

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High-order harmonic generation from two-dimensional materials subjected to intense laser fields

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Abstract

Figures

1. Introduction

2. Methodology

3. Results and Discussions

3.1. The case of linearly-polarized pulse along in-plane direction

3.2. The Case of a linearly-polarized pulse along the out-of-plane direction

3.3. The Case of single circularly-polarized laser pulse

4. Conclusion

Declarations

Author Contribution

References

Additional Declarations

Status:

Version 1