Momentum Matrix Elements 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.

High-throughput discovery of metal oxides with high thermoelectric performance via interpretable feature engineering on small data

In this work, we have proposed a data-driven screening framework combining the interpretable machine learning with high-throughput calculations to identify a series of metal oxides that exhibit both high-temperature tolerance and high power factors. Aiming at the problem of weak generalization ability of small data with power factors at high temperatures, we employ symbolic regression for feature creation which enhances the robustness of the model while preserving the physical meaning of features. 33 candidate metal oxides are finally targeted for high-temperature thermoelectric applications from a pool of 48,694 compounds in the Materials Project database. The Boltzmann transport theory is utilized to perform electrical transport properties calculations at 1,000 K. The relaxation time is approximated by employing constant electron-phonon coupling based on the deformation potential theory. Considering band degeneracy, the electron group velocity is obtained using the momentum matrix element method, yielding 28 materials with power factors greater than 50 μWcm−1K−2. The high-throughput framework we proposed is instrumental in the selection of metal oxides for high-temperature thermoelectric applications. Furthermore, our data-driven analysis and transport calculation suggest that metal oxides rich in elements such as cerium (Ce), tin (Sn), and lead (Pb) tend to exhibit high power factors at high temperatures.

Propagation of the COAM matrix of a twisted Gaussian Schell-model beam

The Coherence-Orbital Angular Momentum (COAM) matrix elements of an optical beam describe its mode-to-mode radial correlations. An analytic expression for the COAM matrix of a Twisted Gaussian Schell-Model (TGSM) beam propagating in a vacuum is derived and analyzed in detail, with the help of the Huygens–Fresnel integral. Additionally, we show how to recover the cross-spectral density of the TGSM beam from its COAM matrix, expressing the result in an elegant symmetric form. Due to the unique properties of the TGSM beam, the analysis of its individual OAM correlations will deliver further insight for their applications to imaging, laser communication, optical tweezers, and other modalities.

Polarization factors and their effects on the rotational gyromagnetic ratio

The effect of the polarizations on the collective gyromagnetic ratio (g R ) has been investigated in detail using the Rotational Invariant Quasiparticle Phonon Nuclear Model (RI-QPNM). The model includes an axially symmetric mean-field potential, monopole pairing, spin-dependent residual interactions, and the restoration forces determined according to Pyatov’s prescription for rotational invariance. The restoration of the rotational symmetry gives a solution at the zero energy associated with the rotational branch of nucleonic motion, allowing us to obtain the g R -factors of the core. The remaining solutions lead to the appearance of configuration mixing in the excitation spectrum of the odd-mass deformed nucleus. The configuration mixing quenches both spin and angular momentum matrix elements and affects the contribution of the odd particle to the g R . It has been demonstrated that the polarization factors associated with the ΔK = 1 matrix elements are essential to achieve quantitative agreement with the experimental data. It has also been shown that the general assumption for core polarization, i.e., is insufficient to explain the experimental data.

A comprehensive study of the velocity, momentum and position matrix elements for Bloch states: Application to a local orbital basis

We present a comprehensive study of the velocity operator, \hat{\boldsymbol{v}}=\frac{i}{\hbar} [\hat{H},\hat{\boldsymbol{r}}]\,𝐯̂=iℏ[Ĥ,𝐫̂], when used in crystalline solids calculations. The velocity operator is key to the evaluation of a number of physical properties and its computation, both from a practical and fundamental perspective, has been a long-standing debate for decades. Our work summarizes the different approaches found in the literature, but never connected before in a comprehensive manner. In particular we show how one can compute the velocity matrix elements following two different routes. One where the commutator is explicitly used and another one where the commutator is avoided by relying on the Berry connection. We work out an expression in the latter scheme to compute velocity matrix elements, generalizing previous results. In addition, we show how this procedure avoids ambiguous mathematical steps and how to properly deal with the two popular gauge choices that coexist in the literature. As an illustration of all this, we present several examples using tight-binding models and local density functional theory calculations, in particular using Gaussian-type localized orbitals as basis sets. We show how the the velocity operator cannot be approximated, in general, by the k-gradient of the Bloch Hamiltonian matrix when a non-orthonormal basis set is used. Finally, we also compare with its real-space evaluation through the identification with the canonical momentum operator when possible. This comparison offers us, in addition, a glimpse of the importance of non-local corrections, which may invalidate the naive momentum-velocity correspondence.

Comparative study of optical characteristics of SQW and DQWs InGaAs/GaAsSb/AlAsSb based compressively strained heterostructures

InP-based InGaAs/GaAsSb/AlAsSb type-II single quantum well (SQW) and double quantum wells (DQWs) heterostructures are compared on the basis of matrix elements, energy dispersion curves, optical gain, and transition energy. The effect on optical gain due to applied external strain and the influence of In-mole fraction and Sb-mole fraction on momentum matrix element and transition wavelength have also been compared between SQW and DQWs heterostructures. The simulation results show that the transition wavelength is located in the short wavelength infrared region (SWIR). SWIR imaging has various applications such as solar cell inspection, surveillance, electronic board inspection, identification and sorting, and more. For the proposed design, a total optical gain of 7450 cm−1 and 13,681 cm−1 is obtained at 0.75 eV and 0.77 eV transition energies for the SQW and DQWs heterostructures. For 1.5 nm QW thickness and 3x1012 cm−2 injected carrier density at 300 K, the transition wavelength increases from 1.67 μm to 1.82 μm and 1.60 μm–1.77 μm for SQW and DQWs heterostructure, respectively on increasing the mole fraction value of In from 0.5 to 0.8 and when the mole fraction value of Sb increases from 0.5 to 0.8, the transition wavelength increases from 1.69 μm to 1.87 μm and 1.62 μm–1.79 μm for SQW and DQWs heterostructure, respectively.

Mitigating Valence Intersubband Absorption in Interband Cascade Lasers

AbstractInterband cascade lasers (ICLs), especially valued for their low power consumption, are particularly appealing for portable, compact, and battery‐driven trace gas sensors. However, their performance notably degrades outside the 3–4 µm region. Here, a solution to overcome current performance limitations is presented. Simulation results based on the eight‐band k· p method employing a generalized momentum matrix element model identify resonant intersubband absorption in the valence band as the causative underlying mechanism. Experimentally, a direct dependence of this resonant absorption on the thickness of the GaInxSb hole‐quantum well (h‐QW) is confirmed. This is reflected in the improvement of the laser's characteristic temperature T0, threshold current density Jth, slope efficiency η, and output power. Extracted waveguide losses from length‐dependent measurements substantiate the key role of the valence intersubband absorption. While the performance improvement is experimentally verified at 4.35 µm, the simulation results additionally show how to mitigate undesired absorption at longer wavelengths, paving the way towards high‐performance continuous‐wave (cw) operation above 6 µm.

Tunability of solar cell with double quantum dot structure

This work introduces a double quantum dot (QD) structure as an intermediate band for high-performance solar cells. The density matrix equations, coupled with the continuity-current equation, are solved numerically to address the interaction between all the states, which is impossible with the rate equation modeling. This modeling allowed us to obtain quantum efficiency (QE). Using the orthogonalized plane wave assumption for WL-QD, the momentum matrix elements of QD-QD, QD-wetting layer (WL), WL-barrier transitions are calculated. Results are simulated in both the excitonic and non-excitonic (electron-hole, eh) cases and exhibit the importance of adding the QD layer.The results confirm the importance of adding the intermediate band (QD layer). Both band-to-band recombination rates and relaxation rates in the DQD structure are modulated according to the energy difference between DQD states. Compared to the eh model, the excitonic model reduces the QD-QD relaxation rates and all band-to-band recombination rates. The quantum efficiency (QE) is attaining at longer recombination and relaxation times that are confirmed depending on the wider energy difference by manipulating the transitions of the DQD structure.

Natural selection rules: new positivity bounds for massive spinning particles

We derive new effective field theory (EFT) positivity bounds on the elastic 2 → 2 scattering amplitudes of massive spinning particles from the standard UV properties of unitarity, causality, locality and Lorentz invariance. By bounding the t derivatives of the amplitude (which can be represented as angular momentum matrix elements) in terms of the total ingoing helicity, we derive stronger unitarity bounds on the s- and u-channel branch cuts which determine the dispersion relation. In contrast to previous positivity bounds, which relate the t-derivative to the forward-limit EFT amplitude with no t derivatives, our bounds establish that the t-derivative alone must be strictly positive for sufficiently large helicities. Consequently, they can provide stronger constraints beyond the forward limit which can be used to constrain dimension-6 interactions with a milder assumption about the high-energy growth of the UV amplitude.

Length-Gauge Optical Matrix Elements in WIEN2k

Hybrid exchange-correlation functionals provide superior electronic structure and optical properties of semiconductors or insulators as compared to semilocal exchange-correlation potentials due to admixing a portion of the non-local exact exchange potential from a Hartree–Fock theory. Since the non-local potential does not commute with the position operator, the momentum matrix elements do not fully capture the oscillator strength, while the length-gauge velocity matrix elements do. So far, length-gauge velocity matrix elements were not accessible in the all-electron full-potential WIEN2k package. We demonstrate the feasibility of computing length-gauge matrix elements in WIEN2k for a hybrid exchange-correlation functional based on a finite difference approach. To illustrate the implementation we determined matrix elements for optical transitions between the conduction and valence bands in GaAs, GaN, (CH3NH3)PbI3 and a monolayer MoS2. The non-locality of the Hartree–Fock exact exchange potential leads to a strong enhancement of the oscillator strength as noticed recently in calculations employing pseudopotentials (Laurien and Rubel: arXiv:2111.14772 (2021)). We obtained an analytical expression for the enhancement factor for the difference in eigenvalues not captured by the kinetic energy. It is expected that these results can also be extended to other non-local potentials, e.g., a many-body GW approximation.

Invariant expansion of the 30-band k·p model and its parameters for III-V compounds

In this paper, we derive a ready-to-use symmetry-invariant expansion of the full-zone 30-band $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ Hamiltonian for the ${T}_{d}$ point group. To find respective parameters, the band structures of III-V materials were calculated within a state-of-the-art density functional theory (DFT) approach and used next as targets to adjust the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ bands. A satisfactory agreement of the $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ model with the DFT band structures, for all tested zinc-blende III-V semiconductors, has been achieved. Values of many of the parameters have not been known so far. We also compare the fitted $k\ifmmode\cdot\else\textperiodcentered\fi{}p$ parameters with the values calculated using momentum matrix elements obtained directly from DFT.

Tuning the responsible parameters for gain characteristics of the novel type-II D-QW (InGaAs) heterostructure

The study made in this paper is directed towards the tuning of the parameters responsible for the shift in optical properties of a novel type-II D-QW (double quantum well) heterostructure having the QW of InGaAs and barrier of GaAsSb. This shift in optical properties has been observed via change in composition (change in mole fraction) of the constituent layers that form the heterostructure. The tunability of such parameters has been studied via computing the envelope functions, localized charge carriers, associated quantized energies and the matrix elements (i.e., optical and momentum matrix elements) which give information about the strength of the optical transitions responsible for optical gain of the quantum well heterostructure. For these computations, a multiband k. p model has been employed. Finally, the computed outcomes clarify that the proposed heterostructure is efficient to emit the radiations of 1.55 μm emission wavelength with high optical gain (6000/cm) and can be utilized in the O, E, and S bands of telecom window in optical communication.

Electronic structure, effective masses and g-factors and strain induced crystal field splitting of WZ-InP by method

In this work, we study the electronic structure of wurtzite (WZ) InP, following a procedure developed earlier by us for WZ-GaN [(2018) J. Phys. Chem. Solids 112, 280] and WZ-ZnO [(2016) Semicond. Sci. Technol. 31, 035018], where the Hamiltonian was diagonalised by taking the conduction band and one valence band at a time. We consider the effects of other valence bands on a given valence band through second order perturbation theory of the Hamiltonian. Valence band dispersions are compared with previous calculations. We notice satisfactory agreement between our results and the previous one. Valence band effective masses are calculated from the valence band dispersions and conduction band effective masses were calculated from standard effective mass formulas. While valence band effective masses compare well with earlier calculations the transverse electronic effective mass is slightly higher compared to the experimental value and other reported values. However, when we use a different set of momentum matrix elements the transverse effective mass agrees well with the experimental value but the longitudinal value becomes slightly less than the experimental value. Thus the calculations are quite sensitive to the change in momentum matrix elements. Additionally, we calculate the effective masses as functions of the wave vector along and Our results for the g-factors are the first of its kind for WZ-InP. These results are quite sensitive to the change of sign of the spin–orbit parameters. The changes are not significant in case of electronic g-factors. But these are quite significant for the transverse valence band effective g-factors. Even change of sign is noticed. Spin–orbit mixing is found to be important and responsible for the observed anisotropy. Electronic g-factors agree well with reported experimental values. We also study the effect of strain on the crystal field splitting energy at the point. Trends obtained as functions of strain match with a previous calculation.

Emission directionality of electronic intraband transitions in stacked quantum dots

We investigate the emission directionality of electronic intraband (intersubband) transitions in stacked coupled quantum dots. Using a well-established eight-band $k \cdot p$ method, we demonstrate that the minor contributions from the valence band mixing into the conduction band govern the polarization and emission directionality of electronic $p$-to-$s$-type intraband transitions. Despite that the contribution from the central-cell part to the momentum matrix element is dominant, we find, the contribution from the matrix elements among the envelope functions cannot be neglected. With the help of an artificial cuboidal quantum dot, we show that the vertical emission from intraband transitions can be tuned via the emitter's vertical aspect ratio. Subsequently, we show that these results can be transferred to more realistic geometries of quantum dots and quantum dot stacks and demonstrate that the vertical emission can be enhanced from $23$ % to $46$ % by increasing the emitter's vertical aspect ratio to the isotropic case with a vertical aspect ratio of 1.0. Therefore, a stacking of a few quantum dots ($\sim$4 for the investigated structures) is already sufficient to improve the vertical radiation significantly. Additionally, we discuss the impact of the number of stacked QDs as well as the effect of the interdot coupling strength on the radiation properties.

High-order harmonic generation from twisted bilayer graphene driven by a midinfrared laser field

Theoretical calculations of high-harmonic generation (HHG) from commensurate twisted bilayer graphene (tBLG) under intense laser fields are performed. The nonlinear electron dynamics in tBLG is considered by solving the Liouville--von Neumann equation for a single-particle density matrix, which combines the full energy bands and momentum matrix elements within the framework of tight-binding approximation. We show that the pump intensity determines the relative magnitude of two components of the harmonic spectrum parallel and perpendicular to driving polarization. The important dependence of HHG on twisted angles and crystal orientations is also presented. Especially in the absence of the relaxation process, the harmonic emission for twisted angles around 10 \ifmmode^\circ\else\textdegree\fi{} exhibits an evident decrease in efficiency per layer compared to monolayer graphene (MLG), which can be interpreted according to Fermi velocity modification. Our calculation also shows that the relative emission efficiency of different harmonic orders between tBLG and MLG contains redundant information on both the dephasing time and an empirical parameter characterizing the decay of the interlayer electron hopping, thus suggesting an all-optical method for the reconstruction of the two parameters. The reconstruction feasibility is successfully demonstrated by a simple optimization algorithm even if considering the possible experimental uncertainty of both driving pulse parameters and high-harmonic signals. Our results show that HHG spectroscopic characteristics in tBLG might serve as a fingerprint to identify the geometric stacking angle and the electronic interaction between adjacent layers, as well as the strong-field laser induced ultrafast dephasing process.

Recombination rates of the double quantum dot solar cell structure

This work proposes a double quantum dot structure as an intermediate-band layer to developing solar cell performance for the first time. The continuity-current equation is coupling with the density matrix equations, which are solved numerically to obtain the quantum efficiency. This modeling will calculate the momentum matrix elements of transitions, consider the orthogonalized plane wave for wetting layer- quantum dot transitions, and covers more characteristics than the rate equations by considering all the possible interactions between the states. The results simulate both the excitonic and nonexcitonic (electron-hole) cases. The work emphasizes adding the double quantum dot layer, which gives flexibility in choosing the energy difference between states controlling the recombination rates. The work refers to the importance of orthogonalized plane-wave in solar cell work. In both models, the band-band recombination is high for slight energy differences between the states, confirms the importance of the double quantum dot system to manipulate transitions between states and obtain higher rates. For the electron-hole model, the leakage is high due to the fast recombination of holes. The discrimination between occupations of states is increasing under the excitonic model due to growing hole occupation. In both models, reducing the quantum dot - quantum dot recombinations and increasing all other recombinations will increase the quantum efficiency. The high quantum dot band-to-band rate increases the quantum efficiency. In the excitonic model, lower rates than in the electron-hole model are enough for high quantum efficiency.

Fine tunability of optical gain characteristics of InGaAs/GaAsSb/InAlAs nano-heterostructure under combined effect of field and temperature

The high optical gain of InGaAs/GaAsSb/InAlAs nano-scale heterostructure has motivated us for its further study under variable temperature and electric field. The presented work is devoted to show the impact of variable temperature and electric field on optical gain characteristics of InGaAs/GaAsSb/InAlAs nano-scale heterostructure which has been assumed to grow on lattice matched GaAs substrate. A six band k.p approach has been utilized to optimize the heterostructure in terms of carrier's wavefunctions, local carriers' density in conduction and valence band and dispersion curves followed by calculating momentum matrix elements and the optical gain spectra. According to the calculation results, the increasing range of temperature (having range of 100–300 K at the interval of 50 K) can control the peak gain significantly (6070 cm−1 to 4470 cm−1) with the red shift in wavelength; while enhanced order of electric field at room temperature can control the gain slightly (4470 cm−1 to ~4150 cm−1) with fine red shift in wavelength. The results achieved in this work will definitely prove the method of controlling the gain characteristics as excellent; and the heterostructure studied will be suitable in design of tunable NIR (near infrared) light sources.

Comprehensive optimization of electronic and optical properties of polar III-nitride laser

A detailed investigation is performed to obtain optimized optical performance of polar III-nitride single quantum well (SQW) semiconductor lasers by band engineering through symmetric and asymmetric tuning of device parameters. Electronic, optical, and threshold characteristics are calculated to analyze the effects of band engineering on device performance. Here, the InN-In0.25Ga0.75 N SQW structure is chosen as a case study. This work shows a significant improvement in optical properties from higher wavefunction overlap integral obtained from band modification due to the tuning which also provides higher momentum matrix elements, spontaneous emission rate, and optical gain, as well. The maximum TE-polarized optical gain for the symmetric structure is found as around 6291 cm−1 for 6 A well width and 6 A barrier width at 710 nm, while for asymmetric laser structure, a maximum TE-polarized optical gain of around 6225 cm−1 at around 800 nm is observed for 6 A well width with asymmetric barrier width of 6 A and 8 A. The optimization is carried out over 500–1600 nm wavelength range to obtain the optimum structure for maximum optical performance at different wavelengths. A time-efficient genetic algorithm-based optimization is also performed that provides the same optimization results in reduced time.

Perturbation approach to ab initio effective mass calculations

A degenerate perturbation k⋅p approach for effective mass calculations is implemented in the all-electron density functional theory (DFT) package WIEN2k. The accuracy is tested on major group IVA, IIIA–VA, and IIB–VIA semiconductor materials. Then, the effective mass in graphene and CuI with defects is presented as illustrative applications. For states with significant Cu-d character additional local orbitals with higher principal quantum numbers (more radial nodes) have to be added to the basis set in order to converge the results of the perturbation theory. Caveats related to a difference between velocity and momentum matrix elements are discussed in the context of application of the method to non-local potentials, such as Hartree–Fock/DFT hybrid functionals and DFT+U.

Ab initio construction of symmetry-adapted k·p Hamiltonians for the electronic structure of semiconductors

While $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonians are frequently used for the description of electronic states in quantum nanostructures, a method is lacking to obtain them in their symmetrized form directly from ab initio band structure calculations of bulk material. We developed a method for obtaining the parameters and the symmetry-adapted form of the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonian from the output of an ab initio band structure calculation. The method consists of (i) evaluation of momentum matrix elements between the wave functions obtained from band structure calculation; (ii) identification of the unitary transformation that transforms these wave functions to the symmetry-adapted basis; (iii) transformation of the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonian to the symmetry-adapted basis. We illustrate the methodology by obtaining $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ Hamiltonians that describe the band structure of zinc-blende CdSe and then we use the Hamiltonians obtained to calculate the electronic states in CdSe quantum wells. Excellent agreement between density functional theory and $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ is obtained for the electronic structure, even for quite thin wells.