Transfer Matrix Formalism Research Articles

Fine Tuning the Light Distribution within the Photoactive Layer by Both Solution‐Processed Anode and Cathode Interlayers for High Performance Polymer Solar Cells

Light management is important for improving light absorption within active layers in polymer solar cells (PSCs). Electrode buffer layers play an important role in modulating the distribution of optical electric filed within the photoactive layer. Herein, the authors employ solution‐processed WOx or ReOx to substitute the acidic poly(3,4‐ethylenedioxythiophene):poly(styrenesulfonate) (PEDOT:PSS) as the anode buffer layer and ZrAcac or HfAcac to replace Mg as the cathode buffer layer. Optical transfer matrix formalism simulation is used to model the absorption spectra, exciton generation rate, and optical electric field distribution of devices. Simulated results shows that stronger absorption, quicker exciton generation rate, and more reasonable photoelectric field distribution can be achieved in the photoactive layer with solution‐processed buffer layer modification, which results in a higher short‐circuit current density (Jsc). Under the guidance of theoretical simulation, the device with architecture of ITO/WOx/PTB7‐Th:PC71BM/HfAcac/Al is optimized. Compared with the traditional PEDOT:PSS‐Mg based device, the Jsc is increased from 16.60 to 18.61 mA cm−2 and the best power conversion efficiency (PCE) is increased from 9.02% to 10.60% for the device with WOx‐HfAcac modification, which is among the best values reported for fullerene‐based PSCs. The good agreement between simulated and experimental results indicates that optical model is a useful tool for device design and optimization.

Exact solution of the two-dimensional scattering problem for a class of δ-function potentials supported on subsets of a line

We use the transfer matrix formulation of scattering theory in two-dimensions (2D) to treat the scattering problem for a potential of the form where , and b are constants, is the Dirac δ function, and g is a real- or complex-valued function. We map this problem to that of and give its exact (nonapproximate) and analytic (closed-form) solution for the following choices of : (i) a linear combination of δ functions, in which case is a finite linear array of 2D δ functions; (ii) a linear combination of with real; (iii) a general periodic function that has the form of a complex Fourier series. In particular we solve the scattering problem for a potential consisting of an infinite linear periodic array of 2D δ functions. We also prove a general theorem that gives a sufficient condition for different choices of to produce the same scattering amplitude within specific ranges of values of the wavelength λ. For example, we show that for arbitrary real and complex parameters, a and , the potentials and have the same scattering amplitude for .

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.

Analysis of a fiber-optic deep-etched silicon Fabry–Perot temperature sensor and modeling its fabrication imperfections

Design and fabrication of an in-plane silicon Fabry–Perot temperature sensor for fiber-optic temperature sensing was reported in our previous work. To fabricate this sensor, deep reactive ion etching process was utilized which is challenging due to the large depth of etching needed for the device. Required optically smooth surfaces and highly vertical sidewalls as well as minimum amount of under-etch are difficult to be achieved in deep-etched structures. Here, the fabrication errors are briefly introduced and thereafter a numerical analysis based on the transfer-matrix formulation for propagation of Gaussian beams across the proposed silicon Fabry–Perot resonator is developed. Finally, the fabricated sensor is modeled and the device performance degradation due to fabrication imperfections is estimated.

Vertical impedance of a pile in layered saturated viscoelastic half-space considering radial inhomogeneity

A simplified analytical approach is presented to investigated the vertical impedance of a pile embedded in layered saturated viscoelastic half-space considering radial inhomogeneity. The vertical dynamic interaction between the soil and pile is simulated by the Beam-on-Dynamic-Winkler-Foundation (BDWF) model. To characterize the variation of the soil properties in the disturbed zone around the pile, an improved complex stiffness transfer model is developed to determine the Winkler moduli based on the wave propagation theory of saturated porous medium. Vertical impedance of the pile head is obtained by solving the differential equation for axial vibration of the pile, combined with transfer-matrix formulations to deal with the layered property of the soil along the pile shaft. The validity and accuracy of the analytical solutions are demonstrated through the comparison examples for the cases of different radial variations of soil properties. Parametric studies are performed to investigate the influences of the Darcy permeability coefficient, width ratio and shear modulus ratio of the disturbed zone on the vertical impedance of the pile. It is revealed that the permeability of the pore fluid has an influence on the impedance of the pile when the Darcy permeability coefficient is large than 1 × 10−3. In addition, the different disturbance degrees of the soil layers along the pile shaft are suggested to take into consideration in analysis of the vertical impedance of the pile embedded in layered saturated viscoelastic half-space.

Quantum formalism for classical statistics

In static classical statistical systems the problem of information transport from a boundary to the bulk finds a simple description in terms of wave functions or density matrices. While the transfer matrix formalism is a type of Heisenberg picture for this problem, we develop here the associated Schrödinger picture that keeps track of the local probabilistic information. The transport of the probabilistic information between neighboring hypersurfaces obeys a linear evolution equation, and therefore the superposition principle for the possible solutions. Operators are associated to local observables, with rules for the computation of expectation values similar to quantum mechanics. We discuss how non-commutativity naturally arises in this setting. Also other features characteristic of quantum mechanics, such as complex structure, change of basis or symmetry transformations, can be found in classical statistics once formulated in terms of wave functions or density matrices. We construct for every quantum system an equivalent classical statistical system, such that time in quantum mechanics corresponds to the location of hypersurfaces in the classical probabilistic ensemble. For suitable choices of local observables in the classical statistical system one can, in principle, compute all expectation values and correlations of observables in the quantum system from the local probabilistic information of the associated classical statistical system. Realizing a static memory material as a quantum simulator for a given quantum system is not a matter of principle, but rather of practical simplicity.

Classical Magnetism and an Integral Formula Involving Modified Bessel Functions

Abstract We study an integral expression that is encountered in some classical spin models of magnetism. The idea is to calculate the key integral that represents the building block for the expression of the partition function of these models. The general calculation allows one to have a better look at the internal structure of the quantity of interest which, in turn, may lead to potentially new useful insights. We find out that application of two different approaches to solve the problem in a general-case scenario leads to an interesting integral formula involving modified Bessel functions of the first kind which appears to be new. We performed Monte Carlo simulations to verify the correctness of the integral formula obtained. Additional numerical integration tests lead to the same result as well. The approach under consideration, when generalized, leads to a linear integral equation that might be of interest to numerical studies of classical spin models of magnetism that rely on the well-established transfer-matrix formalism.

Super periodic potential

In this paper we introduce the concept of super periodic potential (SPP) of arbitrary order n, n∈I+, in one dimension. General theory of wave propagation through SPP of order n is presented and the reflection and transmission coefficients are derived in their closed analytical form by transfer matrix formulation. We present scattering features of super periodic rectangular potential and super periodic delta potential as special cases of SPP. It is found that the symmetric self-similarity is the special case of super periodicity. Thus by identifying a symmetric fractal potential as special cases of SPP, one can obtain the tunnelling amplitude for a particle from such fractal potential. By using the formalism of SPP we obtain the close form expression of tunnelling amplitude of a particle for general Cantor and Smith–Volterra–Cantor potentials.

Optical properties of nucleobase thin films as studied by attenuated total reflection and surface-enhanced Raman spectroscopy

Optical properties of nucleobase thin films were studied by attenuated total reflection (ATR) and surface-enhanced Raman spectroscopy (SERS). Adenine and guanine films were deposited on fused silica and silver at room temperature by thermal evaporation, and the normal dispersion of refractive indices of transparent adenine and guanine films in the visible and near-infrared regions were analyzed. The measured ATR spectra of adenine (guanine) films and numerical simulations by optical transfer matrix formalism demonstrate that the shift of surface plasmon resonance (SPR) wavelength is approximately linearly proportional to the adenine (guanine) film thickness, indicating that SPR can be used for quantitative measurements of biomaterials. The Raman spectra indicated that the adenine (guanine) films can be deposited by thermal evaporation. The adenine (guanine) films on silver exhibited Raman intensity enhancement as compared to those on glass, which was attributed to the SPR effect of silver platform and might play a role as a hot plate for SERS detection of biomaterials.

Integrable Floquet dynamics, generalized exclusion processes and “fused” matrix ansatz

We present a general method for constructing integrable stochastic processes, with two-step discrete time Floquet dynamics, from the transfer matrix formalism. The models can be interpreted as a discrete time parallel update. The method can be applied for both periodic and open boundary conditions. We also show how the stationary distribution can be built as a matrix product state. As an illustration we construct parallel discrete time dynamics associated with the R-matrix of the SSEP and of the ASEP, and provide the associated stationary distributions in a matrix product form. We use this general framework to introduce new integrable generalized exclusion processes, where a fixed number of particles is allowed on each lattice site in opposition to the (single particle) exclusion process models. They are constructed using the fusion procedure of R-matrices (and K-matrices for open boundary conditions) for the SSEP and ASEP. We develop a new method, that we named “fused” matrix ansatz, to build explicitly the stationary distribution in a matrix product form. We use this algebraic structure to compute physical observables such as the correlation functions and the mean particle current.

Electromechanical wave finite element method for interconnected piezoelectric waveguides

A novel finite element (FE) formulation involving external electrical degrees of freedom is proposed for cases dealing with the coupling of mechanical and electrical waveguides through an array of piezoelectric patches. Interconnections between successive periodic unit cells enable the use of the transfer matrix (TM) formalism. It is thus shown that the wave finite element (WFE) method can be applied to electromechanical waveguides. As both mechanical and electrical variables are enclosed in the state vectors, the resulting attenuation and phase constants define waves that propagate in the two domains. For the computation of frequency response functions, it is proposed to employ the Riccati transfer matrix (RTM) method in order to avoid numerical instability. The whole computational method is validated through two examples involving a rod and a beam coupled to passive electrical transmission lines interconnecting a piezoelectric array.

Highly amplified light transmission in a parity-time symmetric multilayered structure.

We propose a parity-time symmetric dielectric-nano-film-dielectric multilayered structure that could facilitate highly amplified transmission of optical power in the infrared spectrum. We have theoretically studied our model using the transfer-matrix formalism. The reflection and the transmission coefficients of the S-matrix are evaluated. The theoretical results are validated by FDTD numerical simulation. We have shown how the thickness of the layers and the gain/loss coefficient of the active layers could generate spectral singularities in the S-matrix and how these singularities could be exploited to achieve amplified transmission of a single wavelength through the structure.

Intersubband surface plasmon polaritons in all-semiconductor planar plasmonic resonators

We theoretically discuss properties of intersubband surface plasmon polaritons (ISPPs) supported by the system consisting of a multiple quantum well (MQW) slab embedded into planar resonator with highly doped semiconducting claddings playing the role of cavity mirrors. Symmetric structures, where the MQW slab occupies the whole space between the claddings and asymmetric structures, where the MQW occupy only half of the space between mirrors, are considered. We focus mainly on the nearly degenerate structures where intersubband frequency is close to frequency of the surface plasmon of the mirrors. The ISPP characteristics are calculated numerically using a semiclassical approach based on the transfer matrix formalism and the effective-medium approximation. The claddings are described by the lossless Drude model. The possibility of engineering the dispersion of the ISPP branches is demonstrated. In particular, for certain parameters of the asymmetric structures we observe the formation of the multimode ISPP branches with two zero group velocity points. We show that the properties of the ISPP branches are reasonably well interpreted employing quasiparticle picture provided that the concept of the mode overlap factor is generalized, taking into account the dispersive character of the mirrors. In addition to this, we demonstrate that the lossless dispersion characteristics of the ISPP branches obtained in the paper are consistent with the angle-resolved reflection-absorption spectra of the GaAlAs-based realistic plasmonic resonators.

Excitation of terahertz modes localized on a layered superconductor: Anomalous dispersion and resonant transmission

We study theoretically the optic transmission through a slab of layered superconductor separated from two dielectric leads by spatial gaps. Based on the transfer matrix formalism along with the Josephson plasma electrodynamic approach, we derive analytic expressions for the transmittance and identify the conditions for the perfect transmission. The special interest of the study is focused on the resonant transmission, which occurs when the wave does not propagate in the spatial gaps. Far from the resonance, the transmittance is exponentially small due to the total internal reflection from the lead-gap interface. However, the excitation of electromagnetic modes localized on the layered superconductor gives rise to a remarkable resonant enhancement of the transmission. Moreover, this phenomenon is significantly modified for the layered superconductors in comparison with usual dielectrics or conductors. The dispersion curves for the modes localized on the layered superconductor are proved to be nonmonotonic, thus resulting in the specific dependence of the transmittance $T$ on the incidence angle $\ensuremath{\theta}$. In particular, we predict the onset of two resonant peaks in the $T(\ensuremath{\theta})$ dependence and their subsequent merge into the broadened single peak with increasing of the wave frequency. Our analytical results are demonstrated by numerical data.

Quantum interference in multi-branched molecules: The exact transfer matrix solutions.

We present a transfer matrix formalism for studying quantum interference in a single molecule electronic system with internal branched structures. Based on the Schrödinger equation with the Bethe ansatz and employing Kirchhoff's rule for quantum wires, we derive a general closed-form expression for the transmission and reflection amplitudes of a two-port quantum network. We show that the transport through a molecule with complex internal structures can be reduced to that of a single two-port scattering unit, which contains all the information of the original composite molecule. Our method allows for the calculation of the transmission coefficient for various types of individual molecular modules giving rise to different resonant transport behaviors such as the Breit-Wigner, Fano, and Mach-Zehnder resonances. As an illustration, we first re-derive the transmittance of the Aharonov-Bohm ring, and then we apply our formulation to N identical parity-time (PT)-symmetric potentials, connected in series as well as in parallel. It is shown that the spectral singularities and PT-symmetric transitions of single scattering cells may be observed in coupled systems. Such transitions may occur at the same or distinct values of the critical parameters, depending on the connection modes under which the scattering objects are coupled.

Dimers on rail yard graphs

We introduce a general model of dimer coverings of certain plane bipartite graphs, which we call rail yard graphs (RYG). The transfer matrices used to compute the partition function are shown to be isomorphic to certain operators arising in the so-called boson–fermion correspondence. This allows to reformulate the RYG dimer model as a Schur process, i.e. as a random sequence of integer partitions subject to some interlacing conditions. Beyond the computation of the partition function, we provide an explicit expression for all correlation functions or, equivalently, for the inverse Kasteleyn matrix of the RYG dimer model. This expression, which is amenable to asymptotic analysis, follows from an exact combinatorial description of the operators localizing dimers in the transfer-matrix formalism, and then a suitable application of Wick’s theorem. Plane partitions, domino tilings of the Aztec diamond, pyramid partitions, and steep tilings arise as particular cases of the RYG dimermodel. For the Aztec diamond, we provide new derivations of the edge-probability generating function, of the biased creation rate, of the inverse Kasteleyn matrix and of the arctic circle theorem.

Transfer Matrix Formalism-Based Analytical Modeling and Performance Evaluation of Perovskite Solar Cells

We present a physics-based analytical model for p-i-n and n-i-p configurations of perovskite solar cell incorporating the effect of surface recombination velocities at the transport layers. We use transfer matrix formalism combined with carrier transport equations to derive a unified optoelectronic expression for wavelength-dependent photocurrent. We establish the accuracy of the proposed model by calculating the current–voltage characteristics of both configurations and comparing them with published experimental data. The root-mean-square error in current–voltage characteristics and relative error in the corresponding efficiencies are found to be < 0.3 mA/cm2 and < 3%, respectively, for all configurations. We further illustrate the applicability of our model by analyzing the effect of changing transport layer material, a new layer insertion in the solar cell structures and/or variation of different layers’ thicknesses. We show that the derived expression overcomes the shortcomings of the existing analytical models of perovskite solar cells, which fail to describe the aforementioned effects.

FTO-free top-illuminated colloidal quantum dot photovoltaics: Enhanced electro-optics in devices

A solar cell device architecture with top-illumination, where the light does not pass through the substrate, is advantageous for many applications. It is also specifically useful for the construction of tandem or multiple junction photovoltaic devices, with illumination through the top solar cell. Here, a top-illuminated colloidal quantum dot solar cell (TI-CQDSC) is demonstrated and compared with a conventional colloidal quantum dot solar cell (C-CQDSC) constructed on a FTO (fluorine doped tin oxide) glass substrate both theoretically and experimentally. The optical electric field distribution in the solar cells with different configuration is simulated using transfer matrix formalism and a more intense optical electric field was observed in TI-CQDSC, leading to a higher exciton generation rate within the colloidal quantum dot solid. The TI-CQDSCs are constructed on both nonconductive glass and flexible substrates, and a maximum power conversion efficiency of 6.4% and 5.6% is achieved, respectively, comparing to that of 5.9% for the C-CQDSC. The improved performance of the top illuminated solar cell is attributed to a combination of enhanced optical electric field intensity in the colloidal quantum dot solid and superior conductivity of the transparent metal film electrode.

Entanglement entropy for 2D gauge theories with matters

We investigate the entanglement entropy in 1+1-dimensional $SU(N)$ gauge theories with various matter fields using the lattice regularization. Here we use extended Hilbert space definition for entanglement entropy, which contains three contributions; (1) classical Shannon entropy associated with superselection sector distribution, where sectors are labelled by irreducible representations of boundary penetrating fluxes, (2) logarithm of the dimensions of their representations, which is associated with "color entanglement", and (3) EPR Bell pairs, which give "genuine" entanglement. We explicitly show that entanglement entropies (1) and (2) above indeed appear for various multiple "meson" states in gauge theories with matter fields. Furthermore, we employ transfer matrix formalism for gauge theory with fundamental matter field and analyze its ground state using hopping parameter expansion (HPE), where the hopping parameter $K$ is roughly the inverse square of the mass for the matter. We evaluate the entanglement entropy for the ground state and show that all (1), (2), (3) above appear in the HPE, though the Bell pair part (3) appears in higher order than (1) and (2) do. With these results, we discuss how the ground state entanglement entropy in the continuum limit can be understood from the lattice ground state obtained in the HPE.

Raman and IR-ATR spectroscopy studies of heteroepitaxial structures with a GaN:C top layer

This work, motivated by the technologically important task of determination of carbon dopant location in the GaN crystal lattice, employed Raman spectroscopy, with both resonant and non-resonant excitation, and infrared (IR) spectroscopy, in the attenuated total reflection (ATR) configuration, to study lattice vibration modes in a set of carbon-doped GaN (GaN:C) epilayers grown by metalorganic vapour phase epitaxy. We analyse Raman and IR-ATR spectra from the point of view of possible effects of the carbon doping, namely: (i) local vibration mode of C atoms in the nitrogen sublattice (whose frequency we theoretically estimate as 768 cm−1 using an isotope defect model), and (ii) shift in the positions of longitudinal modes owing to the phonon–plasmon coupling. We find only indirect hints of the doping effect on the resonant Raman spectra. However, we show theoretically and confirm experimentally that the IR-ATR spectroscopy can be a much more sensitive tool for this purpose, at least for the considered structures. A weak perturbation of the dielectric function of GaN:C, caused by the substitutional carbon impurity, is shown to produce a measurable dip in the ATR reflectivity spectra at ≈770 cm−1 for both p- and s-polarizations. Moreover, it influences a specific (guided-wave type) mode observed at ≈737 cm−1, originating from the GaN layer, which appears in the narrow frequency window where the real parts of the two components of the dielectric tensor of the hexagonal crystal have opposite signs. This interpretation is supported by our modelling of the whole multilayer structure, using a transfer matrix formalism.