T-matrix Formalism Research Articles

Modeling long imperfect SNS junctions and Andreev bound states using two impurities and the T -matrix formalism

We provide a new analytical tool to calculate the energies of Andreev bound states (ABS) in long imperfect SNS junctions, at present these can only be described by numerical tools. We model an NS junction as a delta-function "Andreev" impurity, i.e., a localized potential which scatters an electron into a hole with opposite spin. We show using the scattering matrix formalism that, quite surprisingly, an "Andreev" impurity is equivalent to an NS junction characterized by both Andreev reflection and a finite amount of normal scattering. The ABS energies are then calculated using the T-matrix formalism applied to a system with two Andreev impurities. Our results lie between those for a perfect long SNS junction limit described by the Andreev approximation (ABS energies depend linearly on the phase and are independent of the chemical potential) and the particle-in-the-box limit (bound state energies are independent of the phase and have a linear dependence on the chemical potential). Moreover, we recover a closed-form expression for the ABS energies by expanding around the particle-in-the-box limit.

T-matrix formulation of electromagnetic wave scattering by charged non-spherical scatterers

In this paper, a method is proposed to solve the scattering of electromagnetic waves (EMWs) by a uniformly charged non-spherical scatterer. Based on Waterman's extended boundary method, a transition matrix between coefficient of the scattering field and incident field of a charged non-spherical scatterer is derived, called Tc-matrix. By using Tc-matrix, the scattering fields of charged non-spherical scatterers can be numerically solved. In the case of a charged rotationally symmetric non-spherical scatterer, the analytic mathematical expression of Tc-matrix is obtained, thus the analytical expression of scattering field is obtained. And the extinction characteristics of charged non-spherical scatterers, making use of charged spheroids as examples, are investigated. It is found that absorption is still the dominant mechanism of attenuation when electromagnetic waves travels through small charged spheroidal particles suspended in atmosphere, though the net surface charges can enhance the extinction/scattering/absorption of particle. The enhancement factors of extinction/absorption increases with increasing surface charge density and are much higher than the enhancement factor of scattering. It is shown that the equivalent sphere assumption, which was used in theoretical models of optical or scattering properties of charged or uncharged atmospheric or interstellar particles, could result in large errors, e.g., even over 100% for charged prolate spheroids, in theoretical prediction of absorption/extinction. Moreover, this inaccuracy due to the sphere assumption worsens for spheroids with either low or high aspect ratios h (i.e., prolate spheroid with h << 1, or oblate spheroids with h >> 1).

Surface Green's functions and boundary modes using impurities: Weyl semimetals and topological insulators

In this work we provide a new direct and non-numerical technique to obtain the surface Green's functions for three-dimensional systems. This technique is based on the ideas presented in Phys. Rev. B 100, 081106(R), in which we start with an infinite system and model the boundary using a plane-like infinite-amplitude potential. Such a configuration can be solved exactly using the T-matrix formalism. We apply our method to calculate the surface Green's function and the corresponding Fermi-arc states for Weyl semimetals. We also apply the technique to systems of lower dimensions, such as Kane-Mele and Chern insulator models, to provide a more efficient and non-numerical method to describe the formation of edge states.

Deciphering the near-field response with the far-field wavelength-scanned SERS spectra of 4-mercaptopyridine adsorbed on gold nanocolloidal particles entrapped in Langmuir Reverse Schaefer film of 5CB liquid crystal molecules.

Herein, we have reported the wavelength-scanned (WS) SERS spectra of 4-mercaptopyridine molecules (4-MPy) adsorbed on gold nanoparticles immobilised in a Langmuir Reverse Schaefer (L-RSh) film matrix of 5CB molecules. The WS-SERS spectral profile exhibited an increment in the intensity of the enhanced Raman bands of 4-MPy with an increase in the wavelength of the excitation laser source. The rationale behind the experimental observations was supported by the simulated extinction spectra and the enhancement factor measurements of the modelled systems using the T-matrix formalism. The SERS intensity fluctuations in the band located at 1000 cm-1 for the 4-MPy molecule, as obtained from three different locations in the L-RSh film substrate, were also analyzed. Surprisingly, depending on the spatial locations of the substrates, the intensity fluctuations of the abovementioned band exhibit both Poisson and Gaussian distributions but the maximum number of probe molecules that can reside in the scattering areas under investigation cannot exceed sixteen. These observations suggest that the origin of SERS from single/few molecules or from the ensemble-averaged system cannot be inferred from the statistical distributions of the Raman intensity fluctuations. The present experimental observations also revealed the relation between the near-field plasmonic behaviors of the substrate with the corresponding far-field SERS spectra of the 4-MPy molecule.

Chiral Effective Field Theory Description of Neutrino Nucleon–Nucleon Bremsstrahlung in Supernova Matter

Abstract We revisit the rates of neutrino pair emission and absorption from nucleon–nucleon bremsstrahlung in supernova matter using the T-matrix formalism in the long-wavelength limit. Based on two-body potentials of chiral effective field theory (χEFT), we solve the Lippmann–Schwinger equation for the T-matrix including non-diagonal contributions. We consider final-state Pauli blocking and hence our calculations are valid for nucleons with an arbitrary degree of degeneracy. We also explore the in-medium effects on the T-matrix and find that they are relatively small for supernova matter. We compare our results with one-pion exchange rates, commonly used in supernova simulations, and calculations using an effective on-shell diagonal T-matrix from measured phase shifts. We estimate that multiple-scattering effects and correlations due to the random phase approximation introduce small corrections on top of the T-matrix results at subsaturation densities. A numerical table of the structure function is provided that can be used in supernova simulations.

K_{z} Selective Scattering within Quasiparticle Interference Measurements of FeSe.

Quasiparticle interference (QPI) provides a wealth of information relating to the electronic structure of a material. However, it is often assumed that this information is constrained to two-dimensional electronic states. We show that this is not necessarily the case. For FeSe, a system dominated by surface defects, we show that it is actually all electronic states with negligible group velocity in the z axis that are contained within the experimental data. By using a three-dimensional tight-binding model of FeSe, fit to photoemission measurements, we directly reproduce the experimental QPI scattering dispersion, within a T-matrix formalism, by including both k_{z}=0 and k_{z}=π electronic states. This result unifies both tunnelling based and photoemission based experiments on FeSe and highlights the importance of k_{z} within surface sensitive measurements of QPI.

Resonant Coupling and Gain Singularities in Metal/Dielectric Multishells: Quasi-Static Versus T-Matrix Calculations

We investigate the resonant gain response in doped multishell hybrid nanoparticles made of concentric and alternated doped dielectric and metal shells. In particular, we compare the enhanced extinction properties calculated in a quasi-static approximation with accurate light scattering calculations in the T-matrix formalism. We show that, even for small hybrid particles, a difference in the calculated optoplasmonic mode yields a dramatic change in the resonant coupling with the doped molecular system. Thus, although a simple dipole approach gives a fast qualitative view of the multishell gain-assisted response, a complete light scattering framework is crucial for a quantitative investigation of these hybrid nanosystems.

Hyperbolic hybrid waves and optical topological transitions in few-layer anisotropic metasurfaces

A comprehensive analysis of hybrid TM-TE polarized surface electromagnetic waves supported by different few-layer anisotropic metasurfaces is presented. A generalized 4$\times$4 T-matrix formalism for arbitrary anisotropic 2D layers is developed, from which the general relations for the surface waves dispersions and scattering coefficients are deduced. Using this formalism and the effective conductivity approach, the dispersions and iso-frequency contours (IFCs) topology of the surface waves in various hybrid uniaxial metasurfaces are studied. The existence of hyperbolic plasmon-exciton polaritons in plasmon-exciton hybrids and hyperbolic acoustic waves with strong confinement in both out-of-plane and in-plane directions in uniaxial plasmonic bilayers are predicted. In plasmonic uniaxial metasurfaces on metal films, the elliptic and hyperbolic backward surface waves with negative group velocity are predicted and additional topological transitions in both elliptic and hyperbolic IFCs of the hybrid surface waves are revealed. Ultrathin twisted uniaxial plasmonic bilayers are proposed as systems with the IFCs topological transitions highly sensitive to the layers twist. The developed formalism may become a useful tool in the calculation of multifunctional few-layer metasurfaces or van der Waals heterostructures based on 2D materials with in-plane anisotropy, where the TM-TE polarization mixing must be considered. The predicted effects may open new horizons in the development and applications of planar optical technologies.

Obtaining Majorana and other boundary modes from the metamorphosis of impurity-induced states: Exact solutions via the T-matrix

We provide here a new and exact formalism to describe the formation of end, edge or surface states through the evolution of impurity-induced states. We propose a general algorithm that consists of finding the impurity states via the T-matrix formalism and showing that they evolve into boundary modes when the impurity potential goes to infinity. We apply this technique to obtain Majorana states in 1D and 2D systems described by the Kitaev model with point-like and respectively line-like impurities. We confirm our exact analytical results by a numerical tight-binding approach. We argue that our formalism can be applied to other topological models, as well as to any model exhibiting edge states.

Corroborating the bulk-edge correspondence in weakly interacting one-dimensional topological insulators

We present a Green's function formalism to investigate the topological properties of weakly interacting one-dimensional topological insulators, including the bulk-edge correspondence and the quantum criticality near topological phase transitions, and using interacting Su-Schrieffer-Heeger model as an example. From the many-body spectral function, we find that closing of the bulk gap remains a defining feature even if the topological phase transition is driven by interactions. The existence of edge state in the presence of interactions can be captured by means of a T-matrix formalism combined with Dyson's equation, and the bulk-edge correspondence is shown to be satisfied even in the presence of interactions. The critical exponent of the edge state decay length is shown to be affiliated with the same universality class as the noninteracting limit.

T-matrix formulation of real-space dynamical mean-field theory and the Friedel sum rule for correlated lattice fermions

We formulate real-space dynamical mean-field theory within scattering theory. Thereby the Friedel sum rule is derived for interacting lattice fermions at zero temperature.Graphical abstract

Optical forces in the T-matrix formalism

Optical tweezers are a crucial tool for the manipulation and characterisation, without mechanical contact, of micro- and nanoparticles, ranging from biological components, such as biomolecules, viruses, bacteria, and cells, to nanotubes, nanowires, layered materials, plasmonic nanoparticles, and their composites. Despite the many interdisciplinary applications, only recently it has been possible to develop an accurate theoretical modelling for the mesoscale size range. This goes beyond the strong approximations typically used for the calculation of optical forces on particles much smaller (dipole approximation) or much larger (ray optics) than the wavelength of the trapping light. Among the different methods used to calculate optical forces on model particles, the ones based on the transition matrix (T-matrix) are currently among the most accurate and efficient, particularly when applied to non-spherical particles, both isolated and interacting, or in composite structures. Here, we first give an overview of the theoretical background on optical forces, optomechanics, and T-matrix methods. Then, we focus on calculations of optical trapping on model polystyrene nanowires with the aim to investigate their scaling with nanowire length at the mesoscale. We compare the force constant dependence with approximations at small or large length with respect to the trapping wavelength and with calculations on spheres, pointing out the role of shape.

Ab initio computational analysis of spectral properties of dielectric spheroidal resonators interacting with a subwavelength nanoparticle.

An efficient numerical method for determining the spectral characteristics and spatial distribution of the field of a spheroidal whispering-gallery-mode (WGM) resonator interacting with a dielectric nanoparticle is presented. The developed approach is based on a combination of T-matrix formalism applied to a single resonator with a dipole approximation for the field of the nanoparticle. The method is illustrated by computation of the scattered field of the resonator-particle system illuminated by an incident field in the form of a single WGM mode of TE or TM polarization mimicking the excitation of the resonances by a tapered fiber. Our calculations show that even a very small (less than 0.1%) deviation of the resonator's shape from an ideal sphere renders spherical approximation invalid. They also confirm that analytical resonant approximation for spheroidal resonators developed previously gives a reasonable qualitative description of the spectral characteristics of the resonator-particle system. It was found, however, that corrections to the resonant approximation are significant enough for realistic nominally spherical resonators to be taken into account for accurate analysis of the experimental data.

Chiral optical tweezers for optically active particles in the T-matrix formalism

Modeling optical tweezers in the T-matrix formalism has been of key importance for accurate and efficient calculations of optical forces and their comparison with experiments. Here we extend this formalism to the modeling of chiral optomechanics and optical tweezers where chiral light is used for optical manipulation and trapping of optically active particles. We first use the Bohren decomposition to deal with the light scattering of chiral light on optically active particles. Thus, we show analytically that all the observables (cross sections, asymmetry parameters) are split into a helicity dependent and independent part and study a practical example of a complex resin particle with inner copper-coated stainless steel helices. Then, we apply this chiral T-matrix framework to optical tweezers where a tightly focused chiral field is used to trap an optically active spherical particle, calculate the chiral behaviour of optical trapping stiffnesses and their size scaling, and extend calculations to chiral nanowires and clusters of astrophysical interest. Such general light scattering framework opens perspectives for modeling optical forces on biological materials where optically active amino acids and carbohydrates are present.

Quasistatic limit of the electric-magnetic coupling blocks of the T-matrix for spheroids

The T-matrix formally describes the solution of any electromagnetic scattering problem by a given particle in a given medium at a given wavelength. As such it is commonly used in a number of contexts, for example to predict the orientation-averaged optical properties of non-spherical particles. The T-matrix for electromagnetic scattering can be divided into four blocks corresponding physically to coupling between either magnetic or electric multipolar fields. Analytic expressions were recently derived for the electrostatic limit of the electric-electric T-matrix block T22, of prolate spheroids. In such an electrostatic approximation, all the other blocks were zero. We here analyse the long-wavelength limit for the other blocks (T21, T12, T11) corresponding to electric-magnetic, magnetic-electric, and magnetic-magnetic coupling respectively. Analytic expressions (finite sums) are obtained in the case of spheroidal particles by expressing the fields with solutions to Laplace’s equation, expanding the fields in terms of spheroidal harmonics and applying the boundary conditions. Similar expressions are also presented for the auxiliary matrices in the extended boundary condition method, often used in conjunction with the T-matrix formalism.

Experimental analysis of the response of fiber Bragg grating sensors under non-uniform strain field in a twill woven composite

Fiber Bragg grating optical sensors are nowadays widely employed for strain measurement for structural health monitoring and in experimental mechanics. Compared to other techniques, i.e. electrical strain gauges, fiber Bragg grating offer immunity to electromagnetic interference and allow for long transmission lead lines. Moreover, thanks to multiplexing interrogation, several sensors can be photo-imprinted into a single fiber core allowing for strain evaluation at multiple locations simultaneously. They have high adaptability to composite materials, particularly because it is possible to be embedded into laminates without affecting their strength and stiffness. Fiber Bragg grating strain measurements are based on the detection of the wavelength shift of their peak reflected spectrum. However, subjected to strain gradients, the spectral response of fiber Bragg grating sensors may be distorted and the sharp peak may not be retained. In this work, the response of fiber Bragg grating sensors having different grating lengths and bonded to the surface of a carbon fiber-reinforced twill woven laminate was analyzed. The analysis combined transfer matrix (T-matrix) with digital image correlation methods. Digital image correlation technique was used to capture the non-uniform strain fields in the woven composites and measured strains were employed in T-Matrix algorithm to simulate fiber Bragg grating response. Using this approach, the effect of the length of the fiber Bragg grating on the strain measurement is assessed and results discussed. Moreover, it is shown that T-matrix formulation combined with a non-contact strain field measurement technique, as DIC, is an appropriate technique to simulate the behavior of fiber Bragg grating bonded to composite materials of complex microstructure.

Variability of Microwave Scattering in a Stochastic Ensemble of Measured Rain Drops

While it has been proved that multiple scattering in the microwave frequencies has to be accounted for in precipitation retrieval algorithms, the effects of the random arrangements of drops in space has seldom been investigated. The fact is, a single rain drop size distribution (RDSD) corresponds with many actual 3D distributions of those rain drops and each of those may a priori absorb and scatter radiation in a different way. Each spatial configuration is equivalent to any other in terms of the RDSD function, but not in terms of radiometric characteristics, both near and far from field, because of changes in the relative phases among the particles. Here, using the T-matrix formalism, we investigate the radiometric variability of two ensembles of 50 different 3D, stochastically-derived configurations from two consecutive measured RDSDs with 30 and 31 drops, respectively. The results show that the random distribution of drops in space has a measurable but apparently small effect in the scattering calculations with the exception of the asymmetry factor.

Scattering and absorption in dense discrete random media of irregular particles.

We present an approximate numerical solution for the multiple scattering problem involving densely packed arbitrarily shaped small particles. We define incoherent volume elements that describe the statistics of the random medium and formulate an order-of-scattering solution for the entire random medium. We apply the T-matrix formalism to compute the incoherent interactions of irregular particles in the sequence of scattering events in the Monte Carlo radiative transfer algorithm. The T-matrices for the volume elements of arbitrarily shaped particles are computed by the volume-integral-equation (VIE)-based T-matrix method. We show that the approximate solution is in agreement with the numerically exact VIE solution for a small spherical random medium. Finally, we demonstrate the importance of applying irregular particle shape models in the analysis of multiple scattering by a large random medium of non-spherical particles.

A self-demodulated fiber Bragg grating for investigating impact-induced transient responses of phononic crystal beams

This work presents a fiber Bragg grating (FBG) displacement sensing system to experimentally investigate the dynamic behaviors of phononic crystal (PC) beams through impact-induced early short time or long time transient responses. Based on the couple-mode theory and the optical transfer matrix (T-matrix) formulation, we first show that it is feasible to achieve linear displacement sensing using a single FBG without extra demodulators such as matching gratings. To validate its effectiveness, the proposed self-demodulated FBG system is applied to measure the point-wise transient displacement responses of a cantilever PC beam subjected to steel ball impacts. To demonstrate the transient sensing performance, the measured early short time transient responses are compared with the corresponding ones predicted by the method of reverberation-ray matrix (MRRM) theoretically and the finite element method (FEM) numerically. The excellent agreements verify together the experimental, theoretical and numerical results. Finally, conducting Fast-Fourier transform (FFT) to the early short time or long time transient responses gives the frequency responses that clearly show the existence of the band gaps. In addition to providing a new displacement sensing method using the self-demodulated FBGs, this work also offers another route to investigate the band structures of PC beams through the frequency responses transformed from the early short time transient responses, long before the structural damping becomes dominant.

Impurity-induced states on the surface of a three-dimensional Weyl semimetal

We study theoretically the features of impurity-induced states on the surface of a three-dimensional Weyl semimetal in this work. For calculating the impurity-induced local density of states based on T-matrix formulation, we found that for different Weyl semimetal phases the behaviors of a local impurity exhibit distinguishable prominent features for the surface Fermi arc states. Due to two opposite-directional and -chirality surface currents for a surface, a bound state appears at the unitary limit of scattering intensity near the impurity site. Then the resonance condition for different Weyl semimetal phases and scattering intensity is investigated. Our results can be used to identify distinctive topological phases of Weyl semimetal. Furthermore, the relevance of topological nodal-point and -line systems is discussed. Some relation between our theoretical results and current experimental scheme are also discussed.