Optical Matrix Elements Research Articles

Even-odd-dependent optical transitions of zigzag monolayer black phosphorus nanoribbons

We analytically study the electronic structure and optical properties of zigzag-edged phosphorene nanoribbons (ZPNRs) using the tight-binding Hamiltonian and Kubo formula. By directly solving the discrete Schrodinger equation, we obtain the energy spectra and wavefunctions for N-ZPNR (where TV is the number of transverse zigzag atomic chains) and classify the eigenstates according to the lattice symmetry. Then, we obtain the optical transition selection rule of ZPNRs on the basis of symmetry analysis and analytical expressions of optical transition matrix elements. Under incident light that is linearly polarized along the ribbon, we determine that the optical transition selection rule for N-ZPNR with even- or odd-N is qualitatively different. Specifically, for even-N ZPNRs, the inter- (intra-) band selection rule is An =odd (even) because the parity of the wavefunction corresponding to the n-th subband in the conduction (valence) band is (-1)n[(-1)(n+1)] owing to the presence of C2x symmetry. However, the optical transitions between any subbands are possible owing to the absence of C2x symmetry. Our results provide a further understanding on the electronic states and optical properties of ZPNRs, which are useful for explaining the optical experiment data on ZPNR samples.

Band dispersion and optical gain calculations of staggered type GaAs0.4Sb0·6/In0.7Ga0.3As/GaAs0.4Sb0.6 nano-heterostructure under electric field and [100] strain

In this paper, the numerical calculations for the band dispersion in GaAs0.4Sb0·6/In0.7Ga0.3As/GaAs0.4Sb0.6 staggered nano–scale heterostructure have been carried out for different values of the external electric field (0–200 kV/cm) by solving the 6 × 6 k. p Hamiltonian. In addition, the optical matrix elements have been calculated and their behaviors have been predicted for different values of field, strains and temperature. For the different values of charge carrier's injection, different field strengths and different strains along [100], the optical gain within TE (Transverse Electric) and TM (Transverse Magnetic) modes have been simulated. The maximum optical gain is achieved ~16170/cm at ~2000 nm at room temperature without electric field; whereas under the external electric field of 60 kV/cm at room temperature the optical gain was found to be reduced up to ~11807/cm at ~1955 nm. Further, with the external strain of 8 GPa, the optical gain was found to be 13965/cm at 2079 nm. Moreover, the optical gain was found to shift towards lower values with red shift in wavelength at room temperature within TE and TM modes with increasing external strain along [100] direction. On behalf of the outcomes of the simulation, the modeled heterostructure can be utilized in the design of tunable laser diode operating in MIR (mid infrared region) region.

Cd content dependence of in-plane optical polarization in anisotropically strained c-plane CdZnO/ZnO quantum wells

The in-plane polarization ratio of anisotropically strained c-plane CdZnO/ZnO quantum well (QW) structures was investigated as a function of Cd content. The y-polarized peak intensity is rapidly enhanced with increasing strain relaxation along y-direction while the x-polarized peak intensity is slightly reduced. This is mainly owing to the increase in the y-polarized optical matrix element by a change of the light-hole (LH) envelope wavefunction magnitude. The increasing rate of the y-polarized emission peak by the strain relaxation is shown to be dominant for the QW structure with a relatively high Cd content. As a result, the magnitude of the in-plane polarization ratio rapidly increases with increasing Cd content.

Acceptor-related infrared optical absorption in GaAs/AlGaAs quantum wells

Abstract The results of the theoretical and experimental investigations of optical response of the acceptor centers in narrow GaAs/AlGaAs quantum wells in the infrared spectral range are presented. In the theoretical part, we focused on the probabilities of the photoionization of acceptor centers taking into account the complicated valence band structure and intermixing the light and heavy hole states. We used the finite-difference method to quantize the Luttinger-Kohn Hamiltonian and the decomposition of the impurity potential over the eigenstates of this Hamiltonian with the quantum well profile to obtain the acceptor state energies and wavefunctions in momentum space. The momentum-dependent decomposition coefficients were used to calculate the optical matrix elements of the hole transitions from localized acceptor states to 2D continuum states in dipole approximation. Calculated spectral and polarization dependencies of the infrared optical absorption due to photoionization of acceptors are discussed. Equilibrium absorption spectra of the polarized infrared radiation were measured in a wide temperature range. Spectral positions of the observed absorption peaks are in satisfactory agreement with theoretical calculations.

Optical anisotropy in type-II (110)-oriented GaAsSb/GaAs quantum wells

The optical anisotropy of zinc-blende type-II GaAsSb/GaAs quantum wells (QWs) with (110) crystal orientation was investigated by using the multiband effective-mass theory. The optical matrix element for the y′([11̄0])-polarization rapidly decreases with increasing in-plane wave vector. On the other hand, in the case of the x′([001])-polarization, the optical matrix element is nearly independent of the in-plane wave vector. The optical anisotropy gradually increases with increasing carrier density. This can be explained by the fact that, at a higher carrier density, carriers occupy higher states above k||=0 and the optical matrix element for the x′-polarization becomes larger than that for the y′-polarization at high k||.

Simple theoretical analyses of the Burstein–Moss shift (BMS) revisited for n-GaAs semiconductor with and without band-tailing conditions

The Burstein–Moss shift (BMS) has been revisited for n-type GaAs semiconductor with and without band-tailing conditions. Unlike the earlier reported results, the present theoretical analyses of BMS depicted an oscillatory behavior due to the inclusions of the following relevant factors, viz. (a) $$ \overline{k} $$ dependency of the optical matrix elements, (b) consideration of the Fermi integral (FI) methods for the calculation of Fermi energy in non-degenerately doped cases, to a higher-order terms, and (c) band-tailing effects due to heavy doping in degenerately doped condition. In the present communication, attempt has been made to recapitulate the FI of F1/2(η), for non-degenerately doped semiconductor that showed oscillations due to the presence of confluent hyper-geometric function, obtained by the exact integrations method involved with F1/2(η). However, for the heavy doping with band-tailing cases, the FI is found to be complex function of oscillatory amplitude and a phase angle (α). It must be noted that these oscillatory behaviors, demonstrated in the present paper, seemed to be depicted for the first time in both non-degenerately and degenerately doped semiconductors on the literature in the study of BMS.

First-principles study of the phonon replicas in the photoluminescence spectrum of 4H -SiC

The free-exciton photoluminescence (PL) spectrum of $4H$-SiC exhibits a detailed fine structure due to the different phonons involved in the indirect gap transition. Here a first-principles calculation of these phonons, their symmetry labeling and their contribution to the photoluminescence spectrum is presented. The calculation uses phonons and electron-phonon coupling matrix elements computed via density functional perturbation theory and energy bands and optical phonon matrix elements calculated in density functional theory. The results are in excellent agreement with the experiment for the phonon energies and the polarization dependence of the spectrum. The relative intensities are also in fair agreement if we allow for some phonons within a few meV to be interchanged. There is however a remarkable discrepancy that the experimental spectrum shows a distinct behavior for phonons with energy below $\ensuremath{\sim}55$ meV and above that energy, which is absent in the theory. The experimental PL lines corresponding to phonon energies below 55 meV are about a factor 5--10 smaller in intensity. This is not found in our calculations. The calculations show a similar peak distribution as the experiment in this range but with intensities comparable to those above 55 meV. This indicates that another mechanism outside the scope of the electron-phonon mediated transitions is operative for photon energies of the PL lines closer to the indirect exciton gap than this 55 meV cutoff, which reduces the overall intensity of these lines. We propose that this may result from competition between the phonon-assisted PL and trapping of the electron in the available unoccupied hexagonal site N-donor shallow level at 53-meV binding energy. As part of this study, we also present the phonon dispersions and density of states in $4H$-SiC and the electronic band structure including quasiparticle corrections.

Effects of External Magnetic Fields on Optical Properties of an Oxide Quantum Dot Using the Smorodinsky–Winternitz Potential

The effects of a magnetic field and geometrical confinement on the ground-state energy of an exciton and the optical gain in a CdO/ZnO spherical core/shell quantum dot have been investigated using the Smorodinsky–Winternitz potential. The impact of the dielectric constant discontinuity at the nanostructure boundaries is included, as well as the built-in internal fields comprising the spontaneous and piezoelectric polarizations within the heterostructure. Numerical calculations are carried out to obtain the exciton energy using the variational formalism within the single-band effective-mass approximation, whereas the optical properties are found using the compact density matrix method. The magnetic-field-induced oscillator strength and the transition lifetime of the exciton are investigated as functions of the dot radius in the core/shell quantum dot. The normalized optical matrix elements and the optical gain as a function of the carrier density in the presence of a magnetic field strength are studied. The results reveal that the exciton energy is insensitive to the magnetic field in the strong geometrical confinement region. The optical matrix elements increase with the charge carrier screening for all the magnetic field strengths, and this effect can be applied to improve the optical gain.

Origin of the electromagnetic anisotropy in monolayer black phosphorus

Contrary to empirical observations, lowest-order $k\cdot\hat{p}$ theory predicts that monolayer black phosphorus ("phosphorene") is completely immune to zigzag-polarized optical excitation at the bandgap energy. Using symmetry arguments, we derive a $2\times2$ Hamiltonian under the $k\cdot\hat{p}$ formalism including higher-order corrections, which is used to show that the experimentally-measured band-gap transition with zigzag polarization is dominated by the third order $k\cdot\hat{p}$ perturbation in the interband optical matrix element, whereas the effects of spin-orbit interaction are negligible in this material, consistent with a trivial orbital diamagnetic contribution to the $\textit{g}$-factor.

Optical response of an inverted InAs/GaSb quantum well in an in-plane magnetic field**Project supported by the National Natural Science Foundation of China (Grant Nos. 61076092 and 61290303).

The optical response of an inverted InAs/GaSb quantum well is studied theoretically. The influence of an in-plane magnetic field that is applied parallel to the quantum well is considered. This in-plane magnetic field will induce a dynamical polarization even when the electric field component of the external optical field is parallel to the quantum well. The electron–electron interaction in the quantum well system will lead to the de-polarization effect. This effect is found to be important and is taken into account in the calculation of the optical response. It is found that the main feature in the frequency dependence of the velocity–velocity correlation function remains when the velocity considered is parallel to the in-plane magnetic field. When the direction of the velocity is perpendicular to the in-plane magnetic field, the de-polarization effect will suppress the oscillatory behavior in the corresponding velocity–velocity correlation function. The in-plane magnetic field can change the band structure of the quantum well drastically from a gapped semiconductor to a no-gapped semi-metal, but it is found that the distribution of the velocity matrix elements or the optical transition matrix elements in the wave vector space has the same two-tadpole topology.

Improvement of TE-polarized emission in type-II InAlN–AlGaN/AlGaN quantum well**Project supported by the Talent Introduction Project of Nantong University, China (Grant No. 03081055), the National Natural Science Foundation of China (Grant Nos. 61874168 and 61505090), Top-notch Academic Programs Project of Jiangsu Higher Education Institutions, China (Grant No. PPZY2015B135), the Six Top Talents of Jiangsu Province, China (Grant No. 2016-XCL-052), the Natural

The optical properties of the type-II lineup InxAl1−xN–Al0.59Ga0.41N/Al0.74Ga0.26N quantum well (QW) structures with different In contents are investigated by using the six-by-six K–P method. The type-II lineup structures exhibit the larger product of Fermi–Dirac distribution functions of electron and hole and the approximately equal transverse electric (TE) polarization optical matrix elements ( for the c1–v1 transition. As a result, the peak intensity in the TE polarization spontaneous emission spectrum is improved by 47.45%–53.84% as compared to that of the conventional AlGaN QW structure. In addition, the type-II QW structure with has the largest TE mode peak intensity in the investigated In-content range of 0.13–0.23. It can be attributed to the combined effect of and for the c1–v1, c1–v2, and c1–v3 transitions.

Effects of built-in fields on the optical gain in a CdO/ZnO quantum dot with the Smorodinsky–Winternitz potential

Exciton energy, with and without the inclusion of built-in internal fields, as a function of inner dot radius is investigated in a CdO/ZnO quantum dot with the Smorodinsky–Winternitz potential. The internal fields are owing to spontaneous and piezoelectric polarizations existing between the inner core and outer shell materials in the heterostructure. The dielectric mismatch effect is included. Numerical computations are carried out to obtain the exciton energy employing the variational formulism within the single band effective mass approximation. The energy shift owing to spontaneous and piezoelectric polarizations is found to be 172 meV for 20 Å quantum dot radius. The oscillator strength and the radiative lifetime as a function of dot radius are found. The optical matrix elements and the corresponding gain are studied with the constant carrier density. The optical properties are investigated using density matrix method. It is found that the substantial increase of optical gain is estimated with the inclusion of built-in internal fields.

Evaluation of the optical characteristics of a type-II InAs/GaSb superlattice infrared p–i–n photodetector

We investigated the characteristics of a type-II InAs/GaSb superlattice (SL) with various composition periods based on the k·p band calculation. Considering the application to the infrared photodetectors, the essential material parameters are calculated and evaluated from the electronic band structure as a function of the width of InAs and GaSb layers. Under dark conditions, the smaller effective mass in a short period SL (especially with small GaSb layer width) could result in the reduction of the Shockley–Read–Hall (SRH) leakage current which is proportional to the intrinsic carrier density. However, this merit is also concerned with the increase of the dark current due to the enhanced band-to-band tunneling in high electric field conditions. Moreover, in terms of the photo-absorption property, the smaller effective mass results in the smaller joint density of states, which compensates the contribution of the large optical matrix element to the absorption coefficient in short period SL.

A re-examination of experimental evidence on the spectral dependence of the optical transition matrix element associated with thin-film silicon

We analyze the error associated with the optical transition matrix element spectral dependencies for hydrogenated amorphous silicon and crystalline silicon found by Jackson et al. (Phys Rev B 31:5187–5198, 1985). We find that this error is considerable for photon energies less than 1.5 eV. This suggests that spectral dependencies for the optical transition matrix element, other than those proposed by Jackson et al., are possible for photon energies less than 1.5 eV. With this limitation in mind, we then fit a Lorentzian peak to the crystalline silicon experimental optical transition matrix element result of Jackson et al. Within the framework of the damped harmonic oscillator perspective, plotting the term related to the location of the resonant frequency as a function of the damping coefficient, for a number of thin-film silicon experimental results, we have found a relationship between these model parameters. We suspect that this dependence has important implications for the optical response of various forms of thin-film silicon and related materials.

Impact of the stacking sequence on the bandgap and luminescence properties of bulk, bilayer, and monolayer hexagonal boron nitride

We examine the effects of stacking sequence and number of layers on the electronic and luminescence properties of hexagonal boron nitride (h-BN) structures with first-principles calculations based on density functional and many-body perturbation theory. We explored the variations of the magnitude and character (direct or indirect) of the quasiparticle bandgap and interband optical matrix elements for bulk, bilayer, and monolayer stacking polytypes. Although the fundamental gap for most structures is indirect, phonon-assisted transitions are strong (typically 600 times stronger than bulk Si) and enable efficient deep-ultraviolet (UV) luminescence. The polarization of the emitted light is transverse electric, which facilitates light extraction perpendicularly to the h-BN basal plane. Random stacking in turbostratic BN breaks the crystal symmetry and enables optical transitions across the quasi-direct bandgap, albeit with a weak matrix element. Our results demonstrate that h-BN is a promising material for efficient deep-UV light emitters.

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).

Effects of spin-orbit coupling on the optical response of a material

We investigate the effects of spin-orbit coupling on the optical response of materials. In particular, we study the effects of the commutator between the spin-orbit coupling part of the potential and the position operator on the optical matrix elements. Using a formalism that separates a fullyrelativistic Kleinman-Bylander pseudopotential into the scalar-relativistic and spin-orbit-coupling parts, we calculate the contribution of the commutator arising from spin-orbit coupling to the squared optical matrix elements of isolated atoms, monolayer transition metal dichalcogenides, and topological insulators. In the case of isolated atoms from H ($Z = 1$) to Bi ($Z = 83$), the contribution of spin-orbit coupling to the squared matrix elements can be as large as 14 %. On the other hand, in the cases of monolayer transition metal dichalcogenides and topological insulators, we find that this contribution is less than 1 % and that it is sufficient to calculate the optical matrix elements and subsequent physical quantities without considering the commutator arising from spin-orbit coupling.

Optical transitions and localized edge states in skewed zigzag phosphorene nanoribbons

Using the Tight Binding (TB) parameters extracted from Density Functional Theory (DFT) and Recursive Green's Function method, it is shown that skewed-zigzag black phosphorous (phosphorene) nanoribbons obtain large and tuneable bandgap in response to vertical and transverse electric fields. Depending on the direction of the applied field the mid-gap states could possess the localized or metallic nature i.e. non-zero mid-gap density of states. Adjustability of the bandgap and optical dipole transition matrix elements are explained based on the symmetry of involved band edge states. This promises new electronic and optical devices based on phosphorene nanoribbons.

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.

Ab initio study on localization and finite size effects in the structural, electronic, and optical properties of hydrogenated amorphous silicon

We present a first-principles study of the structural, electronic, and optical properties of hydrogenated amorphous silicon (a-Si:H). To this end, atomic configurations of a-Si:H with 72 and 576 atoms respectively are generated using ab initio molecular dynamics, where the larger structures are defect free, closely matching the experimental situation and enabling the comparison of the electronic and optical properties with experimental results. Density functional theory calculations are applied to both configurations in order to obtain the electronic wave functions. These are analyzed and characterized with respect to their localization and their contribution to the density of states, and are used for calculating ab initio absorption spectra of a-Si:H. The results show that both the size and the defect structure of the configurations modify the electronic and optical properties and in particular the value of the band gap. This value could be improved by calculating quasi-particle (QP) corrections to the single-particle spectra using the G0W0 method. We find that the QP corrections can be described by a set of scissors shift parameters, which can also be used in calculations of larger structures. The analysis of individual contributions to the absorption by evaluating the optical matrix elements indicates that strong localization enhances the optical coupling, but has little effect on the average transition probability 〈|vcv|2〉, for which we find a dependence E2+const on the photon energy E, irrespective of the nature of the initial or final state.