Second-harmonic Generation Susceptibility Research Articles

Stacking Faults Enabled Second Harmonic Generation in Centrosymmetric van der Waals RhI3.

Second harmonic generation (SHG) in van der Waals (vdW) materials has garnered significant attention due to its potential for integrated nonlinear optical and optoelectronic applications. Stacking faults in vdW materials are a typical kind of planar defect that introduces a degree of freedom to modulate the crystal symmetry and resultant SHG response. However, the physical origin and tunability of stacking-fault-governed SHG in vdW materials remain unclear. Here, taking the intrinsically centrosymmetric vdW RhI3 as an example, we theoretically reveal the origin of stacking-fault-governed SHG response, where the SHG response comes from the energetically favorable AC̅ stacking fault of which the electrical transitions along the high-symmetry paths Γ-M and Γ-K in the Brillion zone play the dominant role at 810 nm. Such a stacking-fault-governed SHG response is further confirmed via structural characterizations and SHG measurements. Furthermore, by applying hydrostatic pressure on RhI3, the correlation between structural evolution and SHG response is revealed with SHG enhancement up to 6.9 times, where the decreased electronic transition energies and higher momentum matrix elements due to the stronger interlayer interactions upon compression magnify the SHG susceptibility. This study develops a promising foundation for nonlinear nano-optics applications through the strategic design of stacking faults.

Second-harmonic generation in 2D moiré superlattices composed of bilayer transition metal dichalcogenides.

Moiré superlattices (MSLs) in twisted two-dimensional van der Waals materials feature twist-angle-dependent crystal symmetry and strong optical nonlinearities. By adjusting the twist angle in bilayer van der Waals materials, the second-harmonic generation (SHG) can be controlled. Here, we focus on exploring the electronic and SHG properties of MSLs in 2D bilayer transition metal dichalcogenides (TMDs) with different twist angles through first-principles calculations. We constructed MSL structures of five TMD materials, including three single-phase materials (MoS2, WS2, and MoSe2) and two heterojunctions (MoS2/MoSe2 and MoS2/WS2) with twist angles of 9.4°, 13.2°, 21.8°, 32.2°, and 42.1° without lattice mismatch. Our findings demonstrate a consistent variation in the SHG susceptibility among different TMD MSLs as a response to twist-angle changes. The underlying reason for the twist-angle dependence of SHG is that the twist angle regulates the interlayer coupling strength, affecting the optical band gap of MSLs and subsequently tuning the SHG susceptibility. Through a comparison of the static SHG susceptibility values, we identified the twist angle of 9.4° as the configuration that yields the highest SHG susceptibility (e.g. 358.5 pm V-1 for the 9.4° MoSe2 MSL). This value is even twice that of the monolayer (173.3 pm V-1 for monolayer MoSe2) and AA'-stacked bilayer structures (139.8 pm V-1 for AA' MoSe2). This high SHG susceptibility is attributed to the strong interlayer coupling in the 9.4° MSL, which enhances the valence band energy (contributed by the antibonding orbitals of chalcogen-pz and transition metal-dz2) and consequently leads to a small optical band gap, thus improving the optical transitions. The findings of this study provide a straightforward way to improve the SHG performance of bilayer TMDs and also throw light on the sensitive relationship between the twist angle, band structure and SHG properties of TMD MSLs.

1D Van der Waals Polymers with Nonlinear Optical Performance Approaching Theoretical Upper Limit

AbstractNonlinear optical (NLO) materials are of great importance for applications in lasers, atomic clocks, free‐space communication, etc. Herein, inspired by the recent prediction of excellent second harmonic generation (SHG) performance in van der Waals (vdW) materials with 1D building blocks, 14 new NLO materials are found from 244 bulk crystals constructed with 1D polymers using high‐throughput first‐principles calculations. Nearly half of the new NLO materials exhibit superior NLO performance with SHG susceptibilities approaching the theoretical upper limit. The 2D form of 11 candidates inherits the NLO property covering UV, visible, and infrared regions. Bader charge analysis reveals that the SHG susceptibility is determined by the charge difference of ions on the chains. Finally, it is proposed that superior NLO materials can be found in materials with proper bandgaps and large charge differences on the chains. This work not only screens out candidates with outstanding NLO performance in vdW materials with 1D building blocks but also provides a guideline for the search and design of NLO vdW 1D polymer patterns with excellent NLO properties.

Second Harmonic Generation in Exfoliated Few‐Layer ReS2

AbstractThe second harmonic generation (SHG) is systematically investigated in mechanically exfoliated, few‐layer ReS2 samples. The stacking orders are identified as (AA)nA and (ABy)nA for (2n+1)‐layer samples and (AA)n and (ABy)n for 2n‐layer samples. The layer structure A stands for that of a monolayer, and the layer structure By is mirror to A followed by a one‐tenth unit cell translation along the mirror axis, that is, the b‐axis of the crystal. Only the even‐layer samples with (ABy)n stacking order are SHG active. After carefully extracting the effective bulk susceptibility for SHG, it is found that the spectra are greatly enhanced as the fundamental photon energy matches exciton levels. The polarization dependence is strongly anisotropic, and each susceptibility component shows very different wavelength dependence. The results demonstrate that few‐layer ReS2, with its unique SHG, can be considered as an ideal platform for fundamental research in anisotropic nonlinear optics.

Uniaxial Strain Dependence on Angle-Resolved Optical Second Harmonic Generation from a Few Layers of Indium Selenide

Indium selenide (InSe) is an emerging van der Waals material, which exhibits the potential to serve in excellent electronic and optoelectronic devices. One of the advantages of layered materials is their application to flexible devices. How strain alters the electronic and optical properties is, thus, an important issue. In this work, we experimentally measured the strain dependence on the angle-resolved second harmonic generation (SHG) pattern of a few layers of InSe. We used the exfoliation method to fabricate InSe flakes and measured the SHG images of the flakes with different azimuthal angles. We found the SHG intensity of InSe decreased, while the compressive strain increased. Through first-principles electronic structure calculations, we investigated the strain dependence on SHG susceptibilities and the corresponding angle-resolved SHG pattern. The experimental data could be fitted well by the calculated results using only a fitting parameter. The demonstrated method based on first-principles in this work can be used to quantitatively model the strain-induced angle-resolved SHG patterns in 2D materials. Our obtained results are very useful for the exploration of the physical properties of flexible devices based on 2D materials.

Wavelength dependence of polarization-resolved second harmonic generation from ferroelectric SnS few layers

Two-dimensional group-IV monochalcogenides such as GeS, GeSe, SnS, and SnSe were theoretically predicted as multiferroic materials with two or more ferroic properties. However, most of their bulk crystals are stacked layer by layer with an antiferroelectric manner, which lose the macroscopic in-plane ferroelectricity. In this work, we studied SnS in which the layers are stacked in a ferroelectric manner both experimentally and theoretically. We utilized polarization-resolved second harmonic generation (SHG) microscopy to investigate numerous flakes of ferroelectric SnS few layers on mica substrates. We found the SHG polar patterns dramatically varied in the range of 800 nm and 1000 nm due to the frequency-dependent SHG susceptibilities. First-principles calculations have been performed to study the frequency-dependent and layer-dependent SHG susceptibilities in the ferroelectric SnS with AA and AC stacking orders. The variation trend of calculated SHG polar patterns as a function of frequency agrees well with that of the experimental results. Since polarization-resolved SHG is a noncontact and nondestructive technique to determine the crystal orientation, understanding of its properties is important, especially for monitoring the transition of different ferroic phases.

D6versus d10, Which Is Better for Second Harmonic Generation Susceptibility? A Case Study of K2TGe3Ch8 (T = Fe, Cd; Ch = S, Se)

Two acentric chalcogenide compounds, K2CdGe3S8 and K2CdGe3Se8, were synthesized via conventional high-temperature solid-state reactions. The crystal structures of K2CdGe3S8 and K2CdGe3Se8 were accurately determined by single-crystal X-ray diffraction and crystallize in the K2FeGe3S8 structure type. K2CdGe3S8 is isostructural to K2FeGe3S8 with superior nonlinear optical properties. For the second harmonic generation (SHG) response, K2CdGe3S8 is 18× K2FeGe3S8 for samples of particle size of 38-55 μm. The superior nonlinear optical properties of K2CdGe3S8 over K2FeGe3S8 are mainly contributed by the chemical characteristics of Cd compared with Fe, which are elucidated by nonlinear optical property measurements, electronic structure calculations, and density functional theory calculations. The [CdS4] tetrahedra within K2CdGe3S8 exhibit a higher degree of distortion and larger volume compared to the [FeS4] tetrahedra in K2FeGe3S8. This study possesses a good platform to investigate how d-block elements contribute to the SHG response. The fully occupied d10-elements are better for SHG susceptibility than d6-elements in this study. K2CdGe3S8 is a good candidate as an infrared nonlinear optical material of high SHG response (2.1× AgGaS2, samples of particle size of 200-250 μm), type-I phase-matching capability, high laser damage threshold (6.2× AgGaS2), and good stability.

Study of SHG and LE-O Susceptibilities of InAs Crystal: Linear Absorption Taken into Account

We applied a model involving two coupled anharmonic oscillators (electronic and ionic) to estimate the Second Harmonic Generation (SHG) and Linear Electro-Optic (LE-O) susceptibilities of InAs crystal. The crystal of InAs belongs to III-V group compounds owing cubic zinc-blende-type structure. Linear absorption is considered for the selected spectral region 1250 nm – 390 nm. So, the contribution of the imaginary part of the involved complex linear ionic susceptibility to the resultant SHG and LE-O susceptibilities is taken into account and hence the absolute value of complex-linear ionic susceptibility is used in place of the complex-linear ionic susceptibility in the computation of SHG and LE-O coefficients. All of the four constants (nonlinear strength factors), appearing in the model, are determined with the help of experimental data of SHG susceptibility measured in the selected region of 1076 nm-535 nm. Application of such calculated nonlinear strength factors in the concerned modelled expressions, SHG and LE-O susceptibility coefficients are computed as a function of frequency to illustrate the dispersion in the region of 1250 nm –390 nm.

Defect engineering of second-harmonic generation in nonlinear optical semiconductors

Second-harmonic generation (SHG) plays a key role in developing modern optical devices and quantum optics, as they can expand the wavelength range provided by common laser sources. Conventionally, the modulation of SHG in nonlinear optical (NLO) materials heavily depends on various external effects. In this article, by developing the basic rules that comprise the interplay between defects and the SHG effect, we propose an alternative strategy to intrinsically regulate the SHG of NLO materials via defect engineering, which is feasible via controlling the growth conditions. Using first-principles calculations, we demonstrate that the SHG susceptibility of KBe 2 BO 3 F 2 (KBBF), the only available deep-ultraviolet NLO material with a weak SHG effect, can be dramatically enhanced by nearly one order of magnitude under specific growth conditions. This is comparable to many other famous narrow-band gap NLO crystals. In general, our developed basic rules and material demonstration could be extended to other NLO systems. • The SHG of NLO materials can be intrinsically regulated via defect engineering • The SHG is affected by the in-gap defect level and the concentration of defects • The SHG of KBBF could be dramatically enhanced by nearly one order of magnitude Conventionally, the modulation of second-harmonic generation (SHG) in nonlinear optical (NLO) materials heavily depends on various external factors. By developing basic rules that consider the interplay between defects and the SHG effect, Li et al. propose a strategy to intrinsically regulate the SHG of NLO materials via defect engineering.

Linear and nonlinear optical properties in quantum wells

In this paper, the linear and nonlinear optical properties, especially the optical rectification (OR), the second-harmonic generation (SHG) susceptibility, the third-harmonic generation (THG) susceptibility, the optical absorption coefficients (OACs), and the refractive index changes (RICs) of GaAs-AlGaAs semi-parabolic well (SPW) are studied in detail utilizing the compact density matrix approach and iterative procedure. The states of electron in the SPW with electric field are obtained utilizing the techniques of the displacement harmonic-oscillator and the envelope wave-function. Our results reveal that the confinement frequency, the electric field, the relaxation time, and the incident field strength have considerable influences on the optical properties of the SPW. If it is desired to obtain the OR, SHG, THG, OAC, and RIC with large values, then a relatively weaker incident field strength and confinement frequency, as well as a relatively stronger electric field intensity, should be chosen. If the optical strength is relatively large, then the effect of the nonlinear (third-order) term is important and cannot be neglected when investigating the non-linear optical properties in the SPW. The semiparabolic well is a promising candidate for these properties.

High-pressure control of optical nonlinearity in the polar Weyl semimetal TaAs

The transition metal monopnictide family of Weyl semimetals recently has been shown to exhibit anomalously strong second-order optical nonlinearity, which is theoretically attributed to a highly asymmetric polarization distribution induced by their polar structure. We experimentally test this hypothesis by measuring optical second harmonic generation (SHG) from TaAs across a pressure-tuned polar-to-nonpolar structural phase transition. Despite the high-pressure structure remaining noncentrosymmetric, the SHG yield is reduced by more than 60% by 20 GPa as compared to the ambient pressure value. By examining the pressure dependence of distinct groups of SHG susceptibility tensor elements, we find that the yield is primarily controlled by a single element that governs the response along the polar axis. Our results confirm a connection between the polar axis and the giant optical nonlinearity of Weyl semimetals and demonstrate pressure as a means to tune this effect in situ.

Dramatically Enhanced Second Harmonic Generation in Janus Group‐III Chalcogenide Monolayers

Abstract2D materials are excellent platforms for nonlinear optical (NLO) response, especially second harmonic generation (SHG), due to its large surface to volume ratio and sub‐nanometer thickness. The SHG susceptibility strongly relies on the symmetry of materials. Constructing Janus structures can break the out‐of‐plane mirror symmetry, bringing about asymmetric charge distribution and leading to a built‐in electric field. Consequently, SHG response along the out‐of‐plane direction can be improved. In the current work, combining first‐principles calculations with independent particle approximation, the SHG response of nine Janus group‐III chalcogenide monolayers (M2XX', MM'X2) are systematically evaluated. Both extraordinary in‐plane and out‐of‐plane SHG response are revealed in all the Janus structures. Besides, cation MM'X2 Janus structures exhibit systemically higher SHG response than that of anion M2XX’ due to stronger dipole. Among them, GaInTe2 possesses extremely high out‐of‐plane SHG response with d31 up to 10490.4 pm V−1 at photon energy (PE) of 4.7 eV, enabling promising applications in ultraviolet NLO devices. The SHG intensity polar plots from Janus structures display unusual rotational symmetry at different PEs allowed by the out‐of‐plane SHG components. This work provides theoretical guidelines for further experimental explorations in 2D group‐III monochalcogenide Janus structures and paves the way for their utilization in NLO devices.

Role of interlayer coupling in second harmonic generation in bilayer transition metal dichalcogenides

Little is known about the role of weak interlayer coupling in the second harmonic generation (SHG) effects of two-dimensional van der Waals (vdW) systems. In this article, taking homo-bilayer $MoS_2/MoS_2$ and hetero-bilayer $MoS_2/MoSe_2$ as typical examples, we have systemically investigated their SHG susceptibilities as a function of interlayer hopping strength using first-principles calculations. For the SHG at zero frequency limit of both $MoS_2/MoS_2$ and $MoS_2/MoSe_2$, although the increase of tint can increase the intensities of interlayer optical transitions (IOT), the increased band repulsion around G point can eventually decrease their SHG values; the larger the tint, the smaller the SHG response. For the SHG spectra of $MoS_2/MoSe_2$ in the low photon-energy region, opposite to the $MoS_2/MoS_2$, their peak values are very sensitive to the variable tint, due to the strong $t_{int}$-dependent IOT dominating in the band edge; the larger the tint, the larger the SHG. For the SHG of $MoS_2/MoS_2$ in the high photon-energy region, comparing to the $MoS_2/MoSe_2$, their peak values will decrease in a much more noticeable way as the $t_{int}$ increases, due to the larger reduction of band nesting effect. Our study not only can successfully explain the puzzling experimental observations for the different SHG responses in different bilayer transition-metal dichalcogenides under variable tint, but also may provide a general understanding for designing controllable the SHG effects in the vdW systems.

Strong Second- and Third-Harmonic Generation in 1D Chiral Hybrid Bismuth Halides.

Breaking the symmetry of a crystal structure can enable even-order nonlinear activities, including second-harmonic generation (SHG). The emerging chiral hybrid organic-inorganic metal halides feature unique optical and electronic properties and flexible crystal structures, making them a class of promising nonlinear optical materials. However, their nonlinear response performances are currently inferior to traditional nonlinear crystals, because of the lack of research on resonant enhancement and third-harmonic generation (THG). Herein, we designed chiral hybrid bismuth halides with naturally nonsymmetrical structure to enable SHG. Simultaneously, these chiral compounds preserve 1D crystal structures to create strong free exciton, broad self-trapped exciton (STE), and discrete band energy levels, which facilitate the resonant enhancement of SHG and THG susceptibilities. These new chiral films showcase superior effective SHG susceptibility (χ(2) ∼ 130.5 pm V-1 at an interesting wavelength of 1550 nm), exceeding that of the reference, a commercial LiNbO3 (χ(2) ∼ 83.4 pm V-1) single-crystal film. Furthermore, their THG intensities are even higher than their SHG intensities, with effective THG susceptibility (χ(3)) being ∼9.0 × 106 pm2 V-2 at 1550 nm (37 times that of the reference monolayer WS2). Their high SHG and THG performances indicate the promising future of these 1D chiral hybrid bismuth halides toward nonlinear optical applications.

Inverse problems for nonlinear Maxwell's equations with second harmonic generation

In the current paper we consider an inverse boundary value problem of electromagnetism with nonlinear Second Harmonic Generation (SHG) process. We show the unique determination of the electromagnetic material parameters and the SHG susceptibility parameter of the medium by making electromagnetic measurements on the boundary. We are interested in the case when a frequency is fixed.

Quantitative Polarization-Resolved Second-Harmonic-Generation Microscopy of Glycine Microneedles.

Second-harmonic generation (SHG) is a nonlinear optical process that can provide disease diagnosis through characterization of biological building blocks such as amino acids, peptides, and proteins. The second-order nonlinear susceptibility tensor χ(2) of a material characterizes its tendency to cause SHG. Here, a method for finding the χ(2) elements from polarization-resolved SHG microscopy in transmission mode is presented. The quantitative framework and analytical approach that corrects for micrometer-scale morphology and birefringence enable the determination and comparison of the SHG susceptibility tensors of β- and γ-phase glycine microneedles. The maximum nonlinear susceptibility coefficients are d33 = 15 pm V-1 for the β and d33 = 5.9 pm V-1 for the γ phase. The results demonstrate glycine as a useful biocompatible nonlinear material. This combination of the analytical model and polarization-resolved SHG transmission microscopy is broadly applicable for quantitative SHG material characterization and diagnostic imaging.

Gold‐ and Silver‐Coated Barium Titanate Nanocomposites as Probes for Two‐Photon Multimodal Microspectroscopy

AbstractImproved multiphoton‐excited imaging and microspectroscopy require nanoprobes that can give different nonlinear optical signals. Here, composite nanostructures with a barium titanate core and a plasmonic moiety at their surface are synthesized and characterized. It is found that the core provides a high second‐order nonlinear susceptibility for sensitive second harmonic generation (SHG) imaging in living cells. As a second function in the two‐photon regime, the plasmonic part yields high local fields for resonant and nonresonant surface enhanced hyper Raman scattering (SEHRS). SEHRS complements the one‐photon surface enhanced Raman scattering (SERS) spectra that are also enhanced by the plasmonic shells. Barium titanate silver core–shell (Ag@BaTiO3) composites are specifically suited for SEHRS and SHG excited at 1064 nm, while gold at barium titanate (Au@BaTiO3) nanoparticles can be useful in a combination of SHG and SERS at lower wavelengths, here at 785 nm and 850 nm. The theoretical models show that the optical properties of the BaTiO3 dielectric core depend on probing frequency, shape, size, and plasmonic properties of the surrounding gold nanoparticles so that they can be optimized for a particular type of experiment. These versatile, tunable probes give new opportunities for combined multiphoton probing of morphological structure and chemical properties of biosystems.

Theory of second-order excitonic nonlinearities in transition metal dichalcogenides

Single layers of transition metal dichalcogenides exhibit tightly bound excitons with large oscillator strength making them to promising materials for future optoelectronic applications. Despite that the linear response of these ultrathin materials is extensively studied, a profound understanding of the excitonic second-order nonlinear optical response, including trigonal warping and Berry curvature effects, is lacking so far. To bridge this gap, in this paper we report a microscopic model for the second-order nonlinear susceptibility tensor of these materials. Based on a $kp$ description of the underlying many-particle Hamiltonian we are able to address excitonic features as well as trigonal warping effects, spin-valley coupling, and also Berry curvature to the susceptibility tensor. We present explicit calculation of the latter for the exemplary materials ${\mathrm{MoS}}_{2}$ and ${\mathrm{WS}}_{2}$ on silicon substrate and encapsulated in hBN. For instance we predict resonant susceptibility for second-harmonic generation $0.5--1.3\phantom{\rule{4pt}{0ex}}\mathrm{nm}/\mathrm{V}$ and double-resonant susceptibility $10--30\phantom{\rule{4pt}{0ex}}\mathrm{n}\mathrm{m}/\mathrm{V}$ for the difference and sum-frequency generation processes. Our results are in good agreement with available experimental findings in the literature.

Single- and double-resonant enhancement of second-harmonic generation in asymmetric AlGaN/GaN/AlGaN quantum well heterostructures

The second-harmonic generation susceptibility is theoretically investigated based on the compact density-matrix formalism in two- and three-level AlGaN/GaN systems. The electronic states and their related wave functions were calculated by solving self-consistently the Schrödinger–Poisson equations within the effective mass and Hartree approximations. The presence of spontaneous and piezoelectric polarizations is taken into account in the modeling part. It was revealed from the relevant results that (i) the second-order susceptibility shows a multiple peak structure due to intersubband and their corresponding virtual transitions, (ii) the use of a back doping as well as the optimization of the layer widths in delta-doped AlGaN/GaN asymmetric quantum wells improved considerably the second-harmonic susceptibility, and (iii) the three-level AlGaN/GaN system is the most efficient structure, which generates the second-harmonic with a significant intensity up to 4.36 × 10−5 mV−1. Numerical results of the present work could lay the basis for research in AlGaN-related optoelectronic device applications.

Tailoring the Linear and Second-Order Nonlinear Optical Responses of the Titanium-MIL-125 Metal–Organic Framework through Ligand Functionalization: A First Principles Study

Density functional theory calculations have been performed to investigate the linear and second-order nonlinear optical (NLO) properties of titanium-based MIL-125 metal–organic frameworks in crystalline form, in which the 1,4-benzenedicarboxylate (BDC) linkers are modified by introducing different functional groups or by extending the BDC ligand to contain two (MIL-126) and three (MIL-127) benzene rings. Our results reveal that the functionalization of the BDC linker tends to increase the dielectric constants and the magnitude of birefringence of MIL-125, especially for the aminated derivatives. Correspondingly, the incorporation of a substituent group will improve the phase matching performance of MIL-125. As for the second-harmonic generation (SHG) susceptibility, the SHG activity of the pristine MIL-125 is comparable to KDP, which can be attributed mostly to the contributions of TiO5(OH) octahedra. It is noted that after introducing the substituent group into the BDC linker, the organic part will have a remarkable influence on the SHG intensity. However, the specific effect on the NLO response is dependent on the type of functional group incorporated into the BDC ligand, and only the inclusion of the amine group that is strongly electron-donating can obviously enhance the SHG activity of MIL-125. In addition, MIL-126 and MIL-127 with longer aromatic linking units are not suitable to act as NLO materials due to their poor phase matching abilities, but they are promising candidates for the low dielectric constant materials. The present study can provide theoretical insights to design new second-order NLO materials based on MIL-125.