Momentum Matrix Elements Research Articles

Polarization-resolved analysis of high-order harmonic generation in monolayer MoS2

We employ a theoretical model based on the density-matrix equation in the velocity gauge to calculate high harmonic generation from monolayer MoS2. This approach incorporates the tight-binding model, enabling the full consideration of both crystal symmetry and multiple band effects. In additional to the usual odd harmonics, even harmonics are also presented in the case of observing two different polarization components, which are parallel and perpendicular to the polarization of linearly driving pulses. We detailedly analyze the crystal orientation dependence for the parallel and perpendicular components of both odd- and even-harmonics. It is found that they exhibits different modulation behavior with rotating the crystal orientation. The simulation results capture all important orientation-dependent features observed in the recent experiment, thus demonstrating that the Berry curvature of MoS2 has been appropriately considered in our proposed model. In order to facilitate analysis of the underlying mechanism, we examine the channel current in terms of the contribution from different density-matrix elements, and identify their role in the orientation modulation of high harmonics. We further use simplified one-dimensional integral model to explain the appearance of perpendicular components of even harmonics. Our analysis shows that the multi-band coupling effect is the origin of the parallel even harmonics, while the broken inversion symmetry of phase difference of momentum matrix elements along two orthogonal directions determines the perpendicular even harmonic generation. Additionally, the relationship between the concept of Berry curvature and our theoretical framework is discussed. These demonstrations show that polarization-resolved high harmonics might provide an all-optical way for imaging material’s Berry curvature.

Calculation of optical response functions of dilute-N GaPAsN lattice-matched to Si

Dilute-N GaPAsN alloys have great potential for optoelectronics lattice-matched to Si. However, there is a lack of systematic calculation of the optical response of these alloys. The present paper uses the sp3d5s∗sN tight-binding model to calculate the fullband electronic structure of dilute-N GaPAsN and then calculate the optical response functions considering direct transitions within the electric dipole approximation. Good agreement is obtained for the dielectric function in comparison to available optical data for dilute nitrides. To achieve this, the sp3d5s∗ parameters for GaP and GaAs are optimized for their optical properties in comparison to published data, which are then used as the basis for the sp3d5s∗sN parameters for dilute-N GaPN and GaAsN. The calculated absorption between the valence band and the newly formed lowest conduction band of the dilute nitrides increases as the N fraction increases, in agreement with experiments, mainly due to the net increase in their coupling in the entire Brillouin zone, supported by the calculated momentum matrix element in the present work.

A Plasmonic quantum dot nanolaser: effect of “waveguide Fermi energy” on material gain

This work studies the gain from plasmonic quantum dot (QD) nanolaser. A metal/semiconductor/metal (MSM) structure was considered to attain plasmonic nanocavity with active region contains: QD, wetting layer (WL) and barrier layers. Band alignment between layers was used to predict their parameters. Momentum matrix element for transverse magnetic (TM) mode in QD structure was formulated. Waveguide Fermi energy was introduced and formulated, for the first time, in this work to cover the waveguide contribution (Ag metal layer) in addition to the active region. High net modal gain was obtained when the waveguide Fermi energy was taken into account which means that the increment comes from the material gain not from the confinement factor. The change in waveguide Fermi energy in the valence band explained the high gain, where the valence band QD states are fully occupied referring to an efficient hole contribution.

Giant Net Modal Gain of plasmonic quantum dot nanolaser

This work studies the net modal gain from plasmonic quantum dot (QD) nanolaser. A metal/semiconductor/metal (MSM) structure was considered to attain plasmonic nanocavity with active region contains: QDs, wetting layer (WL) and barrier layers. Band alignment between layers was used to predict their parameters. Momentum matrix element for transverse magnetic (TM) mode in QD structure was formulated. Waveguide Fermi energy was introduced and formulated, for the first time, in this work to cover the waveguide contribution (Ag metal layer) in addition to the active region. Giant net modal gain was obtained when the waveguide Fermi energy was taken into account which means that the increment comes from the material gain not from the confinement factor. The change in waveguide Fermi energy in the valence band explained the high net modal gain, where the valence band QD states are fully occupied referring to an efficient hole contribution.

Plasmonic Quantum Dot Nanolaser: Effect of “Waveguide Fermi Energy”

This study models quantum dot (QD) plasmonic nanolaser. A metal/semiconductor/metal (MSM) structure was considered to attain plasmonic nanocavity. The active region (semiconductor layers) contains the following: QD, wetting layer (WL), and barrier layers. Band alignment between layers was used to predict their parameters. Momentum matrix element for transverse magnetic (TM) mode in QD structure was formulated. Waveguide Fermi energy was introduced and formulated, for the first time, in this work to cover the waveguide contribution (Ag metal layer) in addition to the active region. The high net modal gain was obtained when the waveguide Fermi energy was considered which meant that the increment comes from the material gain, not from the confinement factor. The obtained results were reasoned the high gain due to the change in waveguide Fermi energy in the valence band, where the valence band QD states are fully occupied that are referring to an efficient hole contribution.

Ultrahigh Gain from Plasmonic Quantum Dot Nanolaser

This work studies the gain from quantum dot plasmonic nanolaser. A metal/semiconductor/metal structure was considered to attain plasmonic nanocavity with active region contains: quantum dot, wetting layer and barrier layers. Band alignment between layers was used to predict their parameters. Momentum matrix element for transverse magnetic mode in quantum dot structure was formulated. Waveguide Fermi energy was introduced and formulated, for the first time, in this work to cover the waveguide contribution (Ag metal layer) in addition to the active region. The gain obtained here overcomes the electron scattering losses which promises in high gain, high power and high speed applications. The waveguide Fermi energy goes deep in the valence band which explains the high gain, where it is shown that covering the structure by a metal makes valence band quantum dot states fully occupied which refers to an efficient hole contribution.

Extraction of eight-band k⋅p parameters from empirical pseudopotentials for GeSn

We extract the parameters for the eight-band k⋅p model for Ge1−xSnx (x≤0.15) from the calculation of the nonlocal empirical pseudopotential method with the modified virtual crystal approximation. The atomic pseudopotential form factors of Ge are improved such that the calculated Ge band structure has the commonly accepted bandgap and effective masses. The improved Ge parameters are used in proper interpolation to derive the parameters of GeSn for empirical pseudopotential calculation. The calculated band structures suggest that the Ge1−xSnx alloys exhibit a transition between an indirect bandgap semiconductor and a direct one at Sn composition xc=0.071. From the calculation, we extract the bandgap, the split-off energy, the interband momentum matrix element, and the effective masses of Ge1−xSnx (x≤0.15) as functions of x. From these results, we further derive the parameters used in the eight-band k⋅p model. These parameters are well expressed in quadratic form. The k⋅p model with the extracted parameters can give an interband tunneling current in a “pin” diode that is consistent with the current calculated by the empirical pseudopotential method.

Real time scissor correction in TD-DFT

We demonstrate how the scissor correction to the optical band gap, common in linear-response time-dependent density functional theory (TD-DFT), may be extended to the domain of real-time TD-DFT. This requires modifying both the eigenvalues and momentum matrix elements of the underlying basis set. It provides a simple and computationally economical approach for calculating accurate electron dynamics in solids. We demonstrate the importance of this correction for prototypical semiconductors, diamond and silicon, where the energy absorption in both the linear and non-linear regimes is examined. We also show that for a particular system, ZnSe, using the adiabatic local density approximation together with a scissor correction can be advantageous over other approximations, as the underlying quasi-particle band structure is more accurate.

High pressure affects on optical characteristics of AlGaAs/GaAsP/AlGaAs nano-heterostructure

This paper gives a detailed report on the tunability of optical gain characteristics of type-I n-Al0.45Ga0.55As/GaAs0.84P0.16/p-Al0.45Ga0.55As nano-scale hetrostructure under the external pressures applied along [001] and [100] directions. In order to find the carriers wavefunctions, their localizations and sub-band dispersions in the heterostrcuture, a 6 ×6 K –L Hamiltonian was solved. In addition, the optical and momentum matrix elements were also calculated followed by the calculation of TE and TM optical gain under the applied pressure along [001] and [100] directions. According to the quantitative analysis of the simulated results, it can be suggested that the application of external pressure along [001] direction within TE mode is one of the preferable approaches for the efficient improvement in the gain characteristics of the n-Al0.45Ga0.55As/GaAs0.84P0.16/p-Al0.45Ga0.55As nano-scale hetrostructure based laser performance in the NIR (near ifrared region).

Tight-binding calculations of optical matrix elements for conductivity using nonorthogonal atomic orbitals: Anomalous Hall conductivity in bcc Fe

We present a general formula for the tight-binding representation of momentum matrix elements needed for calculating the conductivity based on the Kubo-Greenwood formula using atomic orbitals, which are in general not orthogonal to other orbitals at different sites. In particular, the position matrix element is demonstrated to be important for delivering the exact momentum matrix element. This general formula, applicable to both orthonormal and non-orthonormal bases, solely needs the information of the position matrix elements and the ingredients that have already contained in the tight-binding representation. We then study the anomalous Hall conductivity in the standard example, ferromagnetic bcc Fe, by a first-principles tight-binding Hamiltonian. By assuming the commutation relation $\hat{\vec{p}} = (i m_e/\hbar) [\hat{H},\hat{\vec{r}}]$, the obtained frequency-dependent Hall conductivity is found to be in good agreement with existing theoretical and experimental results. Better agreement with experiments can be reached by introducing a reasonable bandwidth renormalization, evidencing the strong correlation among 3$d$ orbitals in bcc Fe. Since a tight-binding Hamiltonian can be straightforwardly obtained after finishing a first-principles calculation using atomic basis functions that are generated before the self-consistent calculation, the derived formula is particularly useful for those first-principles calculations.

Analytical study of high absorption region of the absorption edge of a-Si:H using nonlinear regression method

This research is concerned with the re-analysis of optical data (the imaginary part of the dielectric function as a function of photon energy E) of a-Si:H films prepared by Jackson et al. and Ferlauto et al. through using nonlinear regression fitting we estimated the optical energy gap and the deviation from the Tauc model by considering the parameter of energy photon-dependence of the momentum matrix element of the p as a free parameter by assuming that density of states distribution to be a square root function. It is observed for films prepared by Jackson et al. that the value of the parameter p for the photon energy range is is close to the value assumed by the Cody model and the optical gap energy is which is also close to the value of the Cody gap of these films. While for the films prepared by Ferlauto et al. the value of the parameter is equal to and .

Analysis of Biaxially Tensile Strained Ge/SiGe Multiple Quantum Wells for Electro-Absorption Modulators With Low Polarization Sensitivities

We analyze theoretically biaxially tensile strained Ge/Si0.18Ge0.82 multiple quantum wells (MQWs) for electro-absorption modulators with low polarization sensitivities. The difference between TE and TM polarized momentum matrix elements and absorption spectra are discussed. Our calculation indicates that polarization independent electroabsorption can be achieved by introducing 0.78% biaxial tensile strain in the buffer layer. The simulation results show that the maximum polarization independent absorption contrast ratio is 7.6 dB under 0 V/2 V operation at 1485 nm. The absorption contrast ratio is over 6 dB for both TE and TM polarizations with the wavelength ranging from 1467 to 1489 nm. The proposed scheme provides a promising approach to design highly efficient polarization independent Ge/SiGe MQWs electroabsorption modulators for on-chip optical transmission applications.

Design of active region for GaAsP/AlGaAs tensile strain quantum well laser diodes near 800 nm wavelength

As an active region, the tensile strain GaAs1-xPx quantum well plays an important role in the high power semiconductor laser diode with a wavelength of about 800 nm. Accompanied with the improved stability due to the Al-free active region, the GaAs1-xPx quantum well laser also shows a high level of catastrophic optical mirror damage because of the non-absorbing window at the facet, which is formed automatically by the relaxation of the tensile strain GaAs1-xPx material. On the other side, the GaAs1-xPx quantum well laser can provide a transverse magnetic (TM) polarized light source which is important for many solid state laser systems. However, the energy band structure of the tensile strain GaAs1-xPx quantum well is more complicated than that of the compressed or lattice matched quantum well. Although the light hole band is on the top of the heavy hole band for the bulk tensile strain GaAs1-xPx material, the situation may be different from the tensile strain GaAs1-xPx quantum well, in which the first light hole subband lh1 can be either on the top of the first heavy hole subband hh1 or reversed, that will cause the laser to generate either TM or transverse electric (TE) polarized light according to the well structure. So it is meaningful to optimize the tensile strain GaAs1-xPx quantum well structure based on the analysis of the energy band structure. Firstly, according to the 6×6 Luttinger-Kohn theory, the energy band structure of the tensile strain GaAs1-xPx quantum well is calculated by the finite difference method. The relationship between the interband transition energy and the well structure parameters is established. It is found that the well composition x and the well width should increase simultaneously, in order to fix the first subband transition wavelength at about 800 nm. Special attention is paid to the 808 nm quantum well, the valence structures of different well widths are calculated, the detailed analysis of the envelope function shows that the top valence subband is lh1 for wider well width, while it is changed to hh1 for narrower well width. Meanwhile, both the TE and the TM momentum matrix element are calculated as a function of the transverse wave vector for the subband transition from c1 to lh1, lh2, hh1 and hh2, respectively. Further, the threshold optical gains of different well widths are simulated for 808 nm laser diode with the tensile strain GaAs1-xPx quantum well as an active region, the wider well width benefits the TM mode, while the narrower one is favor of TE mode. Finally, according to the threshold carrier density, the relationship between the threshold current density and the well width is analyzed for 808 nm laser diode by considering both the spontaneous and the Auger recombination, an optimum combination of the well width and the well composition exists. For wider well width, the threshold current density will be higher because of the high energy subband carrier filling effect. For narrower well width, the decrease of the optical confinement factor will lead to the increase of threshold current density.

Optical gain tuning within IR region in type-II In0.5Ga0.5As0.8P0.2/GaAs0.5Sb0.5 nano-scale heterostructure under external uniaxial strain

Quaternary and ternary alloy semiconductors offer an extra degree of flexibility in terms of bandgap tuning. Modifications in the wave functions and alterations in optical transitions in quaternary and ternary QW (quantum well) heterostructures due to external uniaxial strain provide valuable insights on the characteristics of the heterostructure. This paper reports the optical gain in strained InGaAsP/GaAsSb type-II QW heterostructure (well width = 20 Å) under external uniaxial strain at room temperature (300 K). The entire heterostructure is supposed to be grown on InP substrate pseudomorphically. Band structure, wave functions, energy dispersion and momentum matrix elements of the heterostructure have been computed. 6 × 6 diagonalised k→·p→ Hamiltonian matrix of the system is evaluated and Luttinger-Kohn model has been applied for the band structure and wavefunction calculations. TE mode optical gain spectrum in the QW-heterostructure under uniaxial strain along [110] is calculated. Optical gain of the heterostructure as a function of 2D carrier density and temperature variation is investigated. The variation of the peak optical gain as a function of As and Sb fractions in InGaAsP as a barrier and GaAsSb as a well respectively is exhibited. For a charge carrier injection of 5×1012/cm2, the TE optical gain is 3952 cm−1 at room temperature under no external uniaxial strain. Significant increase in TE mode optical gain is observed under high external uniaxial strain (1, 5 and 10 GPa) along [110] within IR (Infrared region) region.

First-principles calculations of electronic and optical properties of Fe1−xZnxS2 and Zn1−xMgxO alloys

Electronic properties of Fe1−xZnxS2 and Zn1−xMgxO alloys are performed by using full potential linearized augmented plane wave method (FP-LAPW). Band gap energies have been calculated by using concomitantly LDA with Tran Blaha modified Becke-Johnson (TB-mBJ) potentials. The corrected positions related to valence band maximum (VBM) and conduction band minimum (CBM) have been evaluated by many-body perturbation theory in GW approximation. The electron affinities of these alloys are determined by knowledge the exact position of CBM from the fit of total density of states (TDOS). In the case of 0 at % of (Zn, Mg) alloyed elements, pyrite FeS2 and wurtzite ZnO, respectively, exhibit positive electron affinities of 3.34 eV and 4.34 eV, which are in close with experimental measurements. As for optical properties, the absorption coefficient spectra and refraction index variations are performed from momentum matrix elements and interpreted via the projected density of states. The studied alloys show significant responses in visible range and a blue shift in connection with increasing the alloyed elements.

Investigation of high optical gain in complex type-II InGaAs/InAs/GaAsSb nano-scale heterostructure for MIR applications.

This paper reports a comprehensive theoretical study of W-shaped complex type-II InGaAs/InAs/GaAsSb nano-scale heterostructure consisting of two quantum wells of InAs material using the six-band k.p theory. The entire structure has been supposed to be grown on InP substrate. In order to optimize the optical gain, the probability densities of electrons and holes were optimized in the heterostructure. Following these calculations, dispersion relations for electron and hole energies, and transverse electric and transverse magnetic polarizations dependent dipole matrix elements and momentum matrix elements were calculated and, finally, the optical gain in both polarization modes was calculated. For this optimized complex heterostructure, a very high optical gain of the order of ∼4500 cm-1 in the regime of mid-infrared wavelength ∼3.2 μm has been achieved. The results suggest that the designed nano-heterostructure may be utilized for mid-infrared region (MIR) applications such as chemical and bio-molecular sensing of molecules, for the applications of spectroscopy in the "fingerprint region" of molecular science, and for detection of atmospheric gases that respond to 3.2μm wavelength.

Optical multistability in double quantum dot system: Effect of momentum matrix elements

Optical stability behavior in a unidirectional ring cavity containing a ladder-plus-Y double QD (DQD) structure was modeled in this work using the density matrix theory in parallel with Momentum matrix elements including interdot QD-QD transitions. QD-WL transitions, which were not included in any other bistability work in QD structures, are also modeled here. Pumping optical field reduces the threshold of optical bistability (OB). The coupling field, which dressing DQD states and strengthens Kerr nonlinearity are increased.It is shown that optical multistability (OM) was the figure of merit for the stability behavior in DQD system, in the contrary with other researches, as a result from considering the calculation of all transition momenta, here. Additionally, considering QD-WL transition reduces OM threshold.

Enabling accurate first-principle calculations of electronic properties with a corrected [formula omitted] scheme

A computationally inexpensive k·p-based interpolation scheme is developed that can extend the eigenvalues and momentum matrix elements of a sparsely sampled k-point grid into a densely sampled one. Dense sampling, often required to accurately describe transport and optical properties of bulk materials, can be demanding to compute, for instance, in combination with hybrid functionals in density functional theory (DFT) or with perturbative expansions beyond DFT such as the GW method. The scheme is based on solving the k·p method and extrapolating from multiple reference k-points. It includes a correction term that reduces the number of empty bands needed and ameliorates band discontinuities. We show how the scheme can be used to generate accurate band structures, density of states, and dielectric functions. Several examples are given, using traditional and hybrid functionals, with Si, TiNiSn, and Cu as model materials. We illustrate that d-electron and semi-core states, which are particular challenging for the k·p method, can be handled with the correction scheme if the sparse grid is not too sparse.

Second-order nonlinearity in ladder-plus-Y configuration in double quantum dot structure.

Second-order nonlinear susceptibility (SONS) in a ladder-plus-Y double quantum dot structure was modeled and then studied numerically under the application of an electric field. The density matrix theory was used to formulate the system while the orthogonalized plane waves for wetting layer-quantum dot (WL-QD) were considered to state the momentum matrix elements for this system. It is found that the momenta follow the smallest energy difference between states with an obvious overlap of the mediated states. Since WL-QD momenta are small, neglecting WL gives high SONS. Millimeter waves are predicted, and a huge SONS can be obtained by the application of more optical fields, which is important in medical and biological applications. The possibility of changing light speed between subluminal and superluminal was predicted here. This opens the way for many applications like multichannel waveguide-multichannel quantum information processing, real quality imaging, and temporal clock.

Piezoelectric and deformation potential effects of strain-dependent luminescence in semiconductor quantum well structures

The mechanism of strain-dependent luminescence is important for the rational design of pressure-sensing devices. The interband momentum-matrix element is the key quantity for understanding luminescent phenomena. We analytically solved an infinite quantum well (IQW) model with strain, in the framework of the 6 × 6 k·p Hamiltonian for the valence states, to directly assess the interplay between the spin-orbit coupling and the strain-induced deformation potential for the interband momentum-matrix element. We numerically addressed problems of both the infinite and IQWs with piezoelectric fields to elucidate the effects of the piezoelectric potential and the deformation potential on the strain-dependent luminescence. The experimentally measured photoluminescence variation as a function of pressure can be qualitatively explained by the theoretical results.