Semiconductor Bloch Equations Research Articles

Bloch Gauge symmetry of the semiconductor Bloch equations

Bloch Gauge symmetry of the semiconductor Bloch equations

High harmonic generation from Kagome lattice based on multi-band semiclassical trajectory method

We develop a multi-band semiclassical trajectory (MBSCT) method for studying the high harmonic generation (HHG) from solids, which is fundamentally similar to the Boltzmann equation but describe the electron density distribution in a different way and can simulate the electron transitions between bands, thereby depicting a richer array of physical processes. Compared to other theoretical methods, such as the time-dependent Schrödinger equation, the semiconductor Bloch equation, and time-dependent density functional theory, our MBSCT method avoids issues like massive consumption of computational resources and the need for wave function phase correction. Moreover, we focus on Kagome-type materials to justify the MBSCT method and investigate the influence of flat band on HHG in strong laser fields. The simulated results show that the intensity of certain harmonic orders is suppressed by the flat band, implying harmonic spectroscopy as a potential all-optical approach for characterising nonequilibrium physics of flat-band quantum materials.

High harmonic generation in monolayer indium nitride

In our work, we theoretically investigate high harmonic generation (HHG) in monolayer hexagonal indium nitride (h-InN) based on the semiconductor Bloch equation under strong laser fields. Compared with h-BN, there is no multiplateau in h-InN. This is because the intraband mechanism dominates the total HHG, and the harmonic generated by the intraband current is about three orders of magnitude higher than that generated by the interband polarization. We find that the higher order part of the HHG is mainly supplied by the interband current, which can be analyzed on the basis of the transition dipole moments between the energy bands. In addition, we found that the HHG of h-InN is sensitive to the external strains due to the modified band dispersion in the electronic structures. This study provides a useful reference for understanding the microscopic mechanism of laser-solid interaction.

Sample-orientation effects in solid-state high-harmonic generation: computational study of GaAs

We report on a large-scale simulation study of the sample-orientation effects in solid-state high-harmonic generation, utilizing a fully three-dimensional GaAs model in conjunction with a solver for semiconductor Bloch equations that faithfully reflects the material symmetry and accounts for the optical response from the entire Brillouin zone. The simulations, qualitatively compared to two sets of independent measurements, demonstrate that the orientation-resolved high-harmonic spectra represent an effective tool for rigorous testing of material models. The results are relevant for the future development of all-optical reconstruction of electronic band-structure from the solid-state high-harmonic generation measurements.

Calculation of shift current tensors in two-dimensional transition metal dichalcogenides

We investigate the shift current induced by optical excitation in two-dimensional transition metal dichalcogenides. Our approach is based on a second-order perturbation expansion of the semiconductor Bloch equations which are formulated in the basis of a three-band tight-binding model. From that, shift current tensors have been calculated for several different materials and their dependence on photon energy has been investigated and discussed.

Modulation of High-Order Harmonic Generation from a Monolayer ZnO by Co-rotating Two-Color Circularly Polarized Laser Fields

By numerically solving the two-dimensional semiconductor Bloch equation, we study the high-order harmonic emission of a monolayer ZnO under the driving of co-rotating two-color circularly polarized laser pulses. By changing the relative phase between the fundamental frequency field and the second one, it is found that the harmonic intensity in the platform region can be significantly modulated. In the higher order, the harmonic intensity can be increased by about one order of magnitude. Through time-frequency analysis, it is demonstrated that the emission trajectory of monolayer ZnO can be controlled by the relative phase, and the harmonic enhancement is caused by the second quantum trajectory with the higher emission probability. In addition, near-circularly polarized harmonics can be generated in the co-rotating two-color circularly polarized fields. With the change of the relative phase, the harmonics in the platform region can be altered from left-handed near-circularly polarization to right-handed one. Our results can obtain high-intensity harmonic radiation with an adjustable ellipticity, which provides an opportunity for syntheses of circularly polarized attosecond pulses.

Gauge Invariant Formulation of the Semiconductor Bloch Equations.

We derive gauge invariant semiconductor Bloch equations (GI-SBEs) that contain only gauge invariant band structure; shift vectors, and triple phase products. The validity and utility of the GI-SBEs is demonstrated in intense laser driven solids with broken inversion symmetry and nontrivial topology. The GI-SBEs present a useful platform for modeling and interpreting light-matter interactions in solids, in which the gauge freedom of the Bloch basis functions obscures physics and creates numerical obstacles.

Full-band modeling of AM and FM interband cascade laser frequency combs

A numerical model for the simulation of frequency combs in interband cascade lasers is developed. The approach is based on slowly varying amplitude approximation for electromagnetic field and semiconductor Bloch equations for the electron gas. The electron gas nonequilibrium distribution function is computed explicitly. Effects of both radiative and nonradiative recombination are considered. The model is applied to both amplitude modulated and frequency modulated combs.

Exploring key factors influencing high harmonic generation in monolayer WS2: insights from semi-conductor Bloch equations based on Haldane's model

Exploring key factors influencing high harmonic generation in monolayer WS2: insights from semi-conductor Bloch equations based on Haldane's model

High-order harmonic generation of ZnO crystals in chirped and static electric fields

High harmonic generation in ZnO crystals under chirped single-color field and static electric field are investigated by solving the semiconductor Bloch equation (SBE). It is found that when the chirp pulse is introduced, the interference structure becomes obvious while the harmonic cutoff is not extended. Furthermore, the harmonic efficiency is improved when the static electric field is included. These phenomena are demonstrated by the classical recollision model in real space affected by the waveform of laser field and inversion symmetry. Specifically, the electron motion in k-space shows that the change of waveform and the destruction of the symmetry of the laser field lead to the incomplete X-structure of the crystal-momentum-resolved (k-resolved) inter-band harmonic spectrum. Furthermore, a pre-acceleration process in the solid four-step model is confirmed.

Assessment of tight-binding models for high-harmonic generation in zinc blende materials.

Using a simulator for semiconductor Bloch equations (SBEs) accounting for the entire Brillouin zone, we examine the tight-binding (TB) description for zinc blende structure as a model for high-harmonic generation (HHG). We demonstrate that TB models of GaAs and ZnSe exhibit second-order nonlinear coefficients that compare favorably with measurements. For the higher-order portion of the spectrum, we use the results published by Xia et al. in Opt. Express26, 29393 (2018)10.1364/OE.26.029393 and show that the HHG spectra measured in reflection can be closely reproduced by our simulations free of adjustable parameters. We conclude that despite their relative simplicity, the TB models of GaAs and ZnSe represent useful tools to study both the low- and higher-order harmonic response in realistic simulations.

Graphene valley polarization as a function of carrier-envelope phase in few-cycle laser pulses and its footprints in harmonic signals

We consider coherent dynamics of graphene charged carriers exposed to an intense few-cycle linearly polarized laser pulse. The results, obtained by solving the generalized semiconductor Bloch equations numerically in the Hartree-Fock approximation, taking into account many-body Coulomb interaction, demonstrate strong dependence of the valley polarization on the carrier-envelope phase (CEP), which is interpolated by the simple sinusoidal law. Then we consider harmonic generation in multi-cycle laser field by graphene preliminary exposed to an intense few-cycle laser pulse. We show that the second harmonic's intensity is a robust observable quantity that provides a gauge of CEP for pulse durations up to two optical cycles, corresponding to 40 $\mathrm{fs}$ at the wavelength of 6.2 $\mathrm{\mu m}$.

Coulomb enhancement of high harmonic generation in monolayer transition metal dichalcogenides.

High harmonic generation (HHG) in monolayer MoS2 is studied using fully microscopic many-body models based on the semiconductor Bloch equations and density functional theory. It is shown that Coulomb correlations lead to a dramatic enhancement of HHG. In particular, near the bandgap, enhancements of two orders of magnitude or more are observed for a wide range of excitation wavelengths and intensities. For excitation at excitonic resonances, strong absorption leads to spectrally broad sub-floors of the harmonics that is absent without Coulomb interaction. The widths of these sub-floors depend strongly on the dephasing time for polarizations. For times of the order of 10 fs the broadenings are comparable to the Rabi energies and reach one electronvolt at fields of approximately 50 MV/cm. The intensities of these contributions are approximately four to six orders below the peaks of the harmonics.

Crystal-Momentum-Resolved Contributions to Harmonics in Laser-Driven Graphene

We investigate the crystal-momentum-resolved contributions to high-order harmonic generation in laser-driven graphene by semi-conductor Bloch equations in the velocity gauge. It is shown that each harmonic is generated by electrons with the specific initial crystal momentum. The higher harmonics are primarily contributed by the electrons of larger initial crystal momentum because they possess larger instantaneous energies during the intra-band motion. Particularly, we observe circular interference fringes in the crystal-momentum-resolved harmonics spectrum, which result from the inter-cycle interference of harmonic generation. These circular fringes will disappear if the inter-cycle interference is disrupted by the strong dephasing effect. Our findings can help to better analyze the mechanism of high harmonics in graphene.

Carrier-Envelope Phase-Controlled Residual Current in Semiconductors

With the purpose of achieving current control by using intense laser field manipulation, we investigate the effect of carrier-envelope phase (CEP) on residual current in SiO2 crystals. By solving semiconductor Bloch equations, we found that the CEP can strongly influence the carrier population of the conduction band, which means that it can act as a simple, but useful, tool to control residual current. That is, the resultant asymmetric distribution in the first Brillouin zone gave rise to non-zero residual current. Additionally, we further consider the two-color laser scheme to achieve better control of residual current, showing that asymmetric two-color laser fields can induce the maximum residual current.

High harmonic generation in crystal SiO2 by sub-10-fs laser pulses

The high harmonic generation (HHG) by few-cycle laser pulses is essential for research in strong-field solid-state physics. Through comparison of high harmonic spectra of solids generated by laser pulses with varying durations, we discovered that lasers with good dispersion compensation are capable of producing a broad spectrum of high harmonics. As the pulse duration is further compressed, several interference peaks appear in the broad spectrum. Moreover, we conducted simulations using the semiconductor Bloch equation, considering the effect of Berry curvature, to better understand this process. Our work provides a valuable approach for studying HHG by few-cycle laser pulses in solid materials, expanding the application of HHG in attosecond physics.

Effect of stacking configuration on high harmonic generation from bilayer hexagonal boron nitride.

High harmonic generation from bilayer h-BN materials with different stacking configurations is theoretically investigated by solving the extended multiband semiconductor Bloch equations in strong laser fields. We find that the harmonic intensity of AA'-stacking bilayer h-BN is one order of magnitude higher than that of AA-stacking bilayer h-BN in high energy region. The theoretical analysis shows that with broken mirror symmetry in AA'-stacking, electrons have much more opportunities to transit between each layer. The enhancement in harmonic efficiency originates from additional transition channels of the carriers. Moreover, the harmonic emission can be dynamically manipulated by controlling the carrier envelope phase of the driving laser and the enhanced harmonics can be utilized to achieve single intense attosecond pulse.

Effect of the interference between interband currents on the crystal harmonic spectra

By numerically solving the semiconductor Bloch equation, we study the high-order harmonic emission spectrum of a MgO crystal under the interaction of a linearly polarized laser pulse. Based on two-band and three-band models, by comparing the variation of the harmonic spectrum with the laser peak intensity, it is found that when the laser intensity is lower, the harmonic spectra from both cases are generally consistent above the minimum band gap. However, some harmonic intensities in the three-band model are significantly enhanced or weakened compared to the two-band case when the laser intensity is higher. By analysis of the electron population and the harmonic phase, it is demonstrated that the harmonic enhancement or reduction is caused by the interference between two interband polarization sources, one of which originates from the current between the first conduction band and the valence band and the other of which is from the polarization between the second and the first conduction band. The harmonic interference between different interband currents contains rich dynamic information and has a significant impact on the harmonic structure and yield, which is helpful for further understanding the generation mechanism of crystal harmonics.

Theory for all-optical responses in topological materials: The velocity gauge picture

High Harmonic Generation (HHG), which has been widely used in atomic gas, has recently expanded to solids as a means to study highly nonlinear electronic response in condensed matter and produce coherent high frequency radiation with new properties. Most recently, attention has turned to Topological Materials (TMs) and the use of HHG to characterize topological bands and invariants. Theoretical interpretation of nonlinear electronic response in TMs, however, presents many challenges. In particular, the Bloch wavefunction phase of TMs has undefined points in the Brillouin Zone. This leads to singularities in calculating the inter-band and intra-band transition dipole matrix elements of Semiconductor Bloch Equations (SBEs). Here, we use the laser-electromagnetic velocity gauge ${\boldsymbol p}\cdot {\bf A}(t)$ to numerically integrate the SBEs and treat the singularity in the production of the electrical currents and HHG spectra. We use a prototype of Chern Insulators (CIs), the Haldane model, to demonstrate our approach. We find good qualitative agreement of the velocity gauge compared to the length gauge and the Time-Dependent Density Functional theory in the case of topologically trivial materials such as MoS$_2$. For velocity gauge and length gauge, our two-band Haldane model reproduces key HHG spectra features: ($\textit i$) The selection rules for linear and circular light drivers, ($\textit ii$) The linear cut-off law scaling and ($\textit iii$) The anomalous circular dichroism. We conclude that the velocity-gauge approach captures experimental observations and provides theoretical tools to investigate topological materials.

Two-particle tight-binding description of higher-harmonic generation in semiconductor nanostructures

We develop a quantum mechanical theory to describe the optical response of semiconductor nanostructures with a particular emphasis on higher-order harmonic Generation. Based on a tight-binding approach we take all two-particle correlations into account thus describing the creation, evolution and annihilation of electron and holes. In the limiting case of bulk materials, we obtain the same precision as that achieved by solving the well-established semiconductor Bloch Equations. For semiconducting structures of finite extent, we also incorporate the surrounding space thus enabling a description of electron emission. In addition, we incorporate different relaxation mechanisms such as dephasing and damping of intraband currents. Moreover, the advantage of our description is that, starting from extremely precise material data as e.g., from tight-binding parameters obtained from density-functional-theory calculations, we obtain a numerical description being by far less computationally challenging and resource-demanding as comparable ab-initio approaches, e.g., those based on time-dependent density functional theory.