Parabolic Band Research Articles

High-temperature transport properties of a two-dimensional weakly doped parabolic semiconductor

A version of the Mexican-hat Hamiltonian is used to study high-temperature transport properties of a two-dimensional weakly doped semiconductor with electron-hole symmetric bands. For a finite doping level and a temperature-dependent band gap, we find a closed analytical form of the temperature-dependent chemical potential. The effective concentrations of charge carriers participating in transport coefficients are analyzed in the space spanned by the total electron concentration and temperature. It is shown that these concentrations are the sum of a residual contribution and two thermally activated contributions, with a complicated dependence on temperature. The analytical expression for the Hall coefficient RHis also found. It is argued that it is a non-monotonic function of the doping level with the maximum at the doping nmax that is a linear function of temperature at high enough temperatures. The analysis of the real part of the interband conductivity shows that it is inversely proportional to incoming photon energy at low temperatures and that it is nearly constant over a wide energy range at high temperatures. This results are expected to be of significant importance in understanding transport and optical properties of weakly doped two-dimensional semiconductors with nearly symmetric parabolic bands.&#xD.

Linear and Nonlinear Optical Properties of Symmetric and Asymmetric Double Triangular Quantum Dots Withinside the Presence of Magnetic Field

AbstractLinear and third‐order nonlinear optical absorption coefficients and relative refractive index changes in symmetric and asymmetric double triangular quantum dots are examined theoretically. The dependence of these optical properties on the magnetic field is examined. After calculating energies and wave functions within the effective mass and parabolic band approaches, analytical expressions of linear and nonlinear optical properties are obtained using the compact density matrix approach and iterative method. Numerical calculations are presented for typical GaAs/AlGaAs material. The results show that the magnetic field causes different effects on the and transitions. Moreover, the calculated results also reveal that the resonance frequency and nonlinear contribution are different in symmetric and asymmetric structures. As a result, it is concluded that the magnetic field plays a vital and important role in the electronic and optical properties of the system and can be used to tune the inter‐subband transitions and change the corresponding optical sensitivities.

Enhanced Thermoelectric Performance of Bi0.5Sb1.5Te3 through Precise Pb Doping: Analysis Using the Single Parabolic Band Model

This study investigates the thermoelectric properties of Pb-doped p-type Bi0.5Sb1.5Te3 alloys using the Single Parabolic Band (SPB) model, focusing on optimizing room-temperature performance. We systematically analyze the effects of Pb doping (0, 0.49, 0.65, 0.81, 0.97, and 1.3 at%) on key parameters including density-of-states effective mass (md *), non-degenerate mobility (μ0), weighted mobility (μw), and the thermoelectric quality factor (B-factor) at 323 K. The results reveal that md * reaches a maximum of 1.37 me at 0.97 at% Pb doping, representing a 22.25 % increase over the pristine sample. The highest μ0 of 234.5 cm2 V-1 s-1 is achieved at 0.65 at% Pb, highlighting the complex relationship between doping and carrier mobility. Notably, 0.97 at% Pb doping optimizes thermoelectric performance, yielding the highest μw, power factor, and B-factor. This composition also minimizes lattice thermal conductivity (k1) by 44.93 % compared to the undoped sample, significantly reducing phonon heat conduction. The Callaway-von Baeyer model corroborates these findings, indicating maximized point defect scattering at 0.97 at% Pb. A theoretical peak figure-of-merit (zT) of 1.74 is thus predicted at this doping level, demonstrating a possible substantial enhancement in thermoelectric efficiency upon appropriate carrier concentration tuning. The observed trends in Seebeck coefficient, Hall carrier concentration, and Hall mobility with increasing Pb content provide insights into the underlying mechanisms of performance enhancement. This comprehensive study highlights the critical role of precise Pb doping in optimizing the thermoelectric properties of Bi0.5Sb1.5Te3 alloys for room-temperature applications and establishes a framework for future investigations into similar material systems.

Simplified approach to estimate Lorenz number using experimental Seebeck coefficient for non-parabolic band

Reducing lattice thermal conductivity (κL) is one of the most effective ways for improving thermoelectric properties. However, the extraction of κL from the total measured thermal conductivity can be misleading if the Lorenz (L) number is not estimated correctly. κL is obtained using the Wiedemann–Franz law, which estimates the electronic part of thermal conductivity κe = L σT, where σ and T are electrical conductivity and temperature, respectively. κL is then estimated as κL = κT − L σT. For metallic systems, the Lorenz number has a universal value of 2.44 × 10 −8 WΩ K−2 (degenerate limit), but for non-degenerate semiconductors, the value can deviate significantly for acoustic phonon scattering, the most common scattering mechanism for thermoelectric materials above room temperature. Up until now, L is estimated by solving a series of equations derived from Boltzmann transport equations. For the single parabolic band (SPB) model, an equation was proposed to estimate L directly from the experimental Seebeck coefficient. However, using the SPB model will lead to an overestimation of L in the case of low bandgap semiconductors, which results in an underestimation of κL, sometimes even negative κL. In this article, we propose a simpler equation to estimate L for a non-parabolic band. The experimental Seebeck coefficient, bandgap (Eg), and temperature (T) are the main inputs to the equation, which nearly eliminates the need for solving multiple Fermi integrals besides giving accurate values of L.

Electrical and thermal transport properties of Cr2Se3-Cr2Te3 solid-solution alloy system and estimation of optimal thermoelectric properties

Cr2Se3 is a layered narrow-bandgap semiconductor with a low lattice thermal conductivity of ∼1 W/mK, which can be considered as a potential thermoelectric material. A solid-solution alloy system with Cr2Te3 can be designed to manipulate the electrical and thermal transport properties. In this study, a series of Cr2Se3-Cr2Te3 solid-solution alloys (Cr2(Se1-xTex)3, x = 0, 0.33, 0.5, 0.67, 0.83, and 1) were synthesized, and their electrical and thermal transport properties were investigated. Regarding the electrical transport properties, the power factor (∼0.35 mW/mK2) of Cr2Se3 gradually and significantly decreased as the Te content (x) increased owing to the significant decrease in the Seebeck coefficient that resulted from the large increase in the carrier concentration. Regarding the thermal transport properties, the total thermal conductivity increased with x as a result of the large increase in the electrical conductivity, despite the continuous decrease in the lattice thermal conductivity. Consequently, the thermoelectric figure of merit (zT ∼ 0.20) of Cr2Se3 gradually and significantly decreased to 0.009 for Cr2Te3 at 700 K; thus, no enhancement of zT was observed from the experimental results of simple solid-solution alloying. On the other hand, the thermoelectric quality factors of the solid-solution compositions for x = 0.33–0.83 were found to be enhanced compared to that of the Cr2Se3 sample, implying that zT could be further enhanced by optimizing the carrier concentration. Indeed, enhanced zT values were estimated based on a single parabolic band model for the solid-solution compositions with x = 0.33–0.83 if the carrier concentration was optimized to ∼1019 cm−3, which was found to be owing to the increased nondegenerate and weighted mobility (inferring weaker carrier–phonon interaction). Therefore, it is suggested that the solid-solution alloying approach could provide a feasible strategy to enhance the thermoelectric performance by reducing carrier-phonon interaction when further combined with carrier concentration optimization.

Optimization of the thermoelectric performance of aluminum-doped zinc oxide-graphene heterojunctions under high temperature and high pressure

High-pressure effects can be used to effectively optimize the thermoelectric performance of wide-bandgap oxides. We build upon this finding to show that the synergistic effects of doping with aluminum and graphene composite give excellent thermoelectric performance under high pressure. By employing the parabolic band model, the Pisarenko curves are derived at different pressures, coupled with the band structure of Al-doped ZnO. An analysis indicates that pressure-induced narrowing of the bandgap in Al-doped ZnO reduces the electron excitation energy, thereby optimizing the electrical properties of the samples. The Debye Callaway model is also used to fit the lattice thermal conductivity of the samples, thus enabling an in-depth analysis of the phonon scattering mechanisms. Our analysis suggests that graphene composites significantly enhance point defect scattering and Umklapp scattering, thus increasing the phonon relaxation time and effectively reducing the lattice thermal conductivity of the samples. The improvement in the figure of merit (zT) of ZnO is attributed to the simultaneous enhancement of its electrical properties under high pressure and the synergistic optimization of reducing the lattice thermal conductivity through doping and composites. In this work, we consider a fixed doping ratio of Al and a constant proportion of graphene composite, and analyze the microstructure and thermoelectric properties of 0.1C–ZnAl0.04O (C–ZnAlO) samples synthesized under pressures ranging from 1 to 5 GPa, with the aim of elucidating the influence of pressure and graphene composite on the electro-acoustic transport properties of ZnO. We find that the zT value for a sample synthesized at 5 GPa is increased to 0.27 at 973 K.

Effects of hydrostatic pressure, temperature, and position-dependent mass on the nonlinear optical properties of triple delta-doped GaAs quantum well

In this study, we thoroughly investigate the impacts of hydrostatic pressure, temperature, and position-dependent mass (PDM) on the nonlinear optical properties of asymmetric triple δ-doped GaAs quantum wells. Our analysis covers total optical absorption coefficients, relative refractive index changes, nonlinear optical rectification, second harmonic generation, and third harmonic generation. Initially, we employ PDM to solve the time-independent Schrödinger equation using the diagonalization method under effective mass and parabolic band approaches, considering pressure and temperature dependencies. Utilizing the first four energy eigenvalues and eigenfunctions, we apply the compact density matrix method to compute the system’s nonlinear optical properties numerically. The results indicate a shift in optical property peak positions toward lower (higher) energy spectra with increasing hydrostatic pressure (temperature). Furthermore, the influence of PDM shifts the system’s optical properties toward the higher energy spectrum, resembling the effect of temperature. From an experimental and theoretical perspective, one of the topics that researchers work on most is GaAs-based δ-doped systems (δ-doped heterojunction bipolar transistors, δ-doped field effect transistors, δ-multiple independent gate field effect transistors, etc.). We believe these findings will provide valuable insights for the researchers involved in GaAs-based δ-doped optoelectronic device design.

Low thermal conductivity and enhanced thermoelectric properties of Cu-substituted Bi2-xCuxSr2Co2Oy ceramics

The Bi2Sr2Co2Oy ceramic, known for its stability as a p-type thermoelectric material, has limited thermoelectric performance that necessitates enhancement. In this study, Bi2-xCuxSr2Co2Oy (x = 0, 0.025, 0.05, 0.1, 0.2) ceramic samples were synthesized using the conventional solid-state reaction method. Substitution of Cu2+ for Bi3+ in the samples increased carrier concentration, significantly reducing electrical resistivity. Utilizing the single parabolic band (SPB) model, it was determined that Cu incorporation led to an increased effective mass, thereby enhancing the Seebeck coefficient. Additionally, fine grains and lattice defects induced by Cu2+ substitution effectively scattered phonons, resulting in an ultra-low thermal conductivity of 0.59 Wm−1K−1 at 923 K for the Bi1.8Cu0.2Sr2Co2Oy sample. Consequently, Cu substitution optimized both the electrical and thermal transport properties of Bi2-xCuxSr2Co2Oy samples. The dimensionless thermoelectric figure of merit (ZT) for Bi1.95Cu0.05Sr2Co2Oy reached 0.31 at 923 K, representing a 63 % increase over the pristine Bi2Sr2Co2Oy sample.

Collective Charge Excitations between Moiré Minibands in Twisted WSe_{2} Bilayers Probed with Resonant Inelastic Light Scattering.

We establish low-temperature resonant inelastic light scattering (RILS) spectroscopy as a tool to probe the formation of a series of moiré bands in twisted WSe_{2} bilayers by accessing collective inter-moiré-band excitations (IMBEs). We observe resonances in RILS spectra at energies in agreement with inter-moiré-band transitions obtained from an abinitio based continuum model. Transitions between the first and second moiré band for a twist angle of about 8° are reported and between the first and the third, and higher bands for a twist of about 3°. The signatures from IMBE for the latter highlight a strong departure from parabolic bands with flat minibands exhibiting very high density of states in accord with theory. These observations allow one to quantify the transition energies at the K point where the states relevant for correlation physics are hosted.

Evolution of thermoelectric transport properties of Sb2Te3–Sb2Se3 solid-solution alloys depending on phase formation behavior

Sb2Te3-based alloys exhibit decent thermoelectric transport properties in the mid-temperature range above 550 K. However, the study on Sb2Te3 alloys has been limited in simple doping approach. Herein, evolution in the thermoelectric transport properties associated with the phase formation of solid solution alloys of Sb2(Te1−xSex)3 (x = 0, 0.25, 0.33, 0.5, 0.75, and 1.0) compositions is investigated to widen the strategy to solid solution alloying. A single phase of the Sb2Te3 rhombohedral structure formed from x = 0 to 0.5, whereas mixed phases with an orthorhombic structure of Sb2Se3 was observed at x = 0.75. The electrical conductivity weighted mobility were gradually decreased as Se content increases to x = 0.75. Consequently, the power factor was decreased gradually and significantly to 0.076 mW/mK2 for x = 0.75 (Sb2(Te0·25Se0.75)3) compared with 2.6 mW/mK2 of the pristine sample. The total thermal conductivity of 2.0 W/mK for the Sb2Te3 was significantly reduced gradually to 0.64 W/mK for x = 0.75 owing to simultaneous decrease in electrical and lattice thermal conductivities. The reduction in lattice thermal conductivity is mainly owing to the addition point defect scattering caused by the Se addition. Nevertheless, thermoelectric figure of merit zT of Sb2Te3 is gradually decreased from 0.64 to 0.11 for x = 0.75 at 650 K due to the degradation in electrical transport properties. Furthermore, the theoretical zT was estimated for all samples with varying carrier concentration based on single parabolic band model.

Enhanced thermoelectric performance of Cr-doped ZnSb through lattice thermal conductivity reduction below the phonon glass limit

Zinc antimonide (ZnSb) is a promising thermoelectric material due to its economic viability, elemental abundance, and superior phase stability compared to other Zn-Sb compounds. However, its widespread application is hindered by a low figure-of-merit (zT). Extensive research has explored doping, alloying, and nanostructuring to improve zT. This study investigates the impact of Cr doping in CryZn1-ySb (y = 0.0 – 0.03). Cr doping with y = 0.01 significantly reduces lattice thermal conductivity below the phonon glass limit. Debye-Callaway model calculations suggest a combined effect of point defects and decreased grain size caused by Cr incorporation. This reduction translates to enhanced thermoelectric performance, with the y = 0.01 sample exhibiting a 70 % improvement in zT at 673 K, reaching ∼0.67. Exceeding the Cr substitution limit leads to the formation of a detrimental secondary CrSb2 phase, increasing thermal conductivity. The Single Parabolic Band model calculations predict further zT enhancement in y = 0.01 sample to ∼0.2 at 300 K through optimized carrier concentration (307 % improvement in zT). These findings demonstrate the potential for significant zT improvement in ZnSb via defect engineering and carrier concentration tuning.

The effect of hydrostatic pressure, temperature and impurity factor on linear and nonlinear optical properties of InxGa1-xAs tuned quantum dot with modified Gaussian potential under the action of a vertical magnetic field

This study presents a detailed theoretical analysis of the effects of hydrostatic pressure, temperature, impurity factors, and external electric fields on the optical properties of InxGa1-xAs tuned quantum dot. A uniform magnetic field, directed perpendicular to the quantum dot plane, is applied. The system is excited using a monochromatic electromagnetic field, considering electron intersubband transitions. Energy levels and wave functions are determined using the parabolic band approximation and effective mass approximation. The linear and nonlinear optical absorption coefficients and refractive index changes are computed using the compact-density matrix approach and the iterative method. Numerical results indicate that the peaks of the linear and nonlinear optical absorption coefficients and refractive index changes shift towards lower energies (red shift) and higher energies (blue shift) under varying hydrostatic pressure, temperature, and impurity factor strength. Additionally, the peak values of the absorption coefficients and refractive index changes increase or decrease based on these parameter variations. These results have significant implications for the design and optimization of quantum dot-based optoelectronic devices.

Modulation of nonlinear optical rectification, second, and third harmonic generation coefficients in n-type quadruple δ-doped GaAs quantum wells under external fields

The paper theoretically explores the variations in nonlinear optical rectification (NOR), second (SHG), and third harmonic generation (THG) coefficients of an n-type quadruple δ-doped GaAs quantum well in the presence of electric, magnetic, and THz laser fields. The energies of subbands and associated wave functions are derived through the solution of the Schrödinger equation, utilizing effective mass and parabolic band approximations. The findings indicate that the applied external electric and magnetic fields significantly influence both the magnitudes and the positions of the peaks in NOR, SHG, and THG concerning the incident photon energies. In addition to the influences of electric and magnetic fields, subjecting the system to an intense laser field results in noticeable changes in the THG, SHG, and NOR coefficients. These numerical findings regarding the optical properties of quadruple δ-doped quantum wells in the presence of external fields could offer valuable guidance for experimental studies.

Nonlinear optical properties of GaAs/AlGaAs asymmetric triple quantum wells designed for quantum cascade lasers: Effect of externally applied electric and magnetic fields

We explore the impact of external fields on the absorption coefficients and relative refractive index changes in a quantum cascade laser profile-like heterostructure that consists of a triple asymmetric quantum well. Firstly, we solve numerically the time-independent Schrödinger equation by using the diagonalization method under the effective mass and parabolic band approaches, to obtain the energy levels and wavefunctions. Then, we evaluate the optical coefficients from the analytical expressions derived from the compact density matrix method. The obtained results put forth that the resonant peaks of optical coefficients shift to higher (lower) energies by increasing the value of the electric (magnetic) field. These results indicate that it would be possible to adjust the optical properties of these structures to the desired range with external fields. As a result, we hope that new optoelectronic devices based on asymmetric triple quantum wells can be developed using the features found here.

Multimode Friedel oscillations in monolayer and bilayer graphene

This study systematically explores the influence of charged impurities on static screening in monolayer graphene and extends the investigation to AA-stacked and AB-stacked bilayer graphene (BLG). Applying the random phase approximation (RPA), monolayer graphene displays unique beating Friedel oscillations (FOs) in inter-valley and intra-valley channels. Shifting to BLG, the study emphasizes layer-specific responses on each layer by considering self-consistent field interactions between layers. It also explores the derived multimode FOs, elucidating distinctions from monolayer behavior. In AA-stacked BLG, distinct metallic screening behaviors are revealed, uncovering unique oscillatory patterns in induced charge density, providing insights into static Coulomb scattering effects between two Dirac cones. The exploration extends to AB-stacked BLG, unveiling layer-specific responses of parabolic bands in multimode FOs with increasing Fermi energy. This comprehensive investigation, integrating RPA considerations, significantly advances our understanding of layer-dependent static screening in the broader context of FOs in graphene, providing valuable contributions to the field of condensed matter physics.

Estimation of Maximum <i>zT</i> in Cu<sub>3</sub>SbSe<sub>4</sub> for Different Starting Materials Content

Cu<sub>3</sub>SbSe<sub>4</sub> is considered a promising thermoelectric material because of its large effective mass and low thermal conductivity, originating from its unique lattice structure. However, Cu<sub>3</sub>SbSe<sub>4</sub> has intrinsically low carrier concentration and relatively high electric resistance which limit performance. Recently, a <i>zT</i> improvement in Cu<sub>3</sub>SbSe<sub>4</sub> was reported where doping/precipitation is controlled by changing the content of the starting materials. However, the effect of these changes in starting content on electronic band structures has not been studied. Here, we investigate how the change in starting materials content (<i>x</i> varying from 6 to 20) affects band parameters like density-of-states effective mass (<i>m<sub>d</sub></i> *), non-degenerate mobility (<i>μ<sub>0</sub></i>), weighted mobility (<i>μ<sub>W</sub></i>), and B-factor using the Single Parabolic Band (SPB) model. For <i>x</i> greater than 8, precipitation of the secondary phase (CuSe) was observed, and the band parameters changed differently for <i>x</i> greater than 8. The <i>m<sub>d</sub></i> * increases up to <i>x</i> = 8 and then rapidly decreases for <i>x</i> > 8. For <i>μ<sub>0</sub></i>, an overall decrease is observed for increasing <i>x</i>, but the rate of decrease is suppressed for <i>x</i> > 8. The <i>μ<sub>W</sub></i> reaches the maximum at <i>x</i> = 8. As <i>x</i> increases, the experimental lattice thermal conductivity also increases, especially for <i>x</i> > 8. Therefore, the B-factor, which is directly related to the theoretical maximum zT, becomes maximum at = 8. Hence the SPB model predicts a maximum <i>zT</i> of 0.0484 for <i>x</i> = 8 at 300 K, which is 15.5% higher than the experimental <i>zT</i> of 0.0419, which can be achieved by tuning the Hall carrier concentration to 4.44 × 10<sup>19</sup> cm<sup>-3</sup>.

Transport properties in ABC-ABA-ABC trilayer graphene junctions

Trilayer graphene (TLG) consists of three layers of graphene arranged in a particular stacking order. In the case of ABC-ABA-ABC stacking, the layers are arranged in an A-B-C sequence, followed by an A-B-A sequence, and again an A-B-C sequence. This stacking arrangement introduces specific electronic properties and band structures due to the different stacking configurations. We focus on elucidating the transport properties of a p-n-p junction formed with ABC-ABA-ABC stacking TLG. Employing the transfer matrix method and considering continuity conditions at the junction boundaries, we establish transmission and reflection probabilities, along with conductance. Notably, electron transport through the ABC-ABA-ABC junction exhibits Klein tunneling, resulting in substantial conductance even in the absence of a potential barrier V 0. This effect arises from the effective barrier induced by our specific stacking, facilitating the passage of a maximal number of electrons. However, the presence of V 0 diminishes Klein tunneling, leading to conductance minima. Furthermore, our findings highlight that interlayer bias δ induces a hybridization of the linear and parabolic bands of ABA-TLG within the junction, reducing resonances. In cases where δ ≠ 0 and V 0 ≠ 0, we observe a suppression of the gap, contrary to the results obtained in ABC tunneling studies where a gap exists.

Observation of parabolic electron bands on superconductor LaRu2As2

Abstract Ru-based superconductor LaRu2As2 has been discovered exhibiting the highest critical temperature of ∼ 7.8 K among iron-free transition metal pnictides with the ThCr2Si2-type crystal structure. However, microscopic research on this novel superconducting material is still lacking. Here, we utilize scanning tunneling microscopy/spectroscopy to uncover the superconductivity and surface structure of LaRu2As2. Two distinct terminating surfaces are identified on the cleaved crystals, namely, the As surface and the La surface. Atomic missing line defects are observed on the La surface. Both surfaces exhibit a superconducting gap of ∼ 1.0 meV. By employing quasiparticle interference techniques, we observe standing wave patterns near the line defects on the La atomic plane. These patterns are attributed to quasiparticle scattering from two electron type parabolic bands.

Thermoelectric optimization using first principles calculation and single parabolic band model: a case of Ca0.5La0.5−x Bi x MnO3 (x = 0, 0.25)

Conducting optimization calculations for thermoelectric performance can be beneficial in guiding the direction of further experimental work. In our study, we utilize a combination of the first principle, Boltzmann transport and restructured single parabolic band model to investigate the half-doped semiconductors based on manganite. Ca0.5La0.5−x Bi x MnO3 (x = 0, 0.25) as samples shows the power factor (PF) optimum value of 30% and 69% for x = 0 and 0.25, respectively at a temperature of 800 K. In addition, both samples show two to three orders of magnitude smaller lattice thermal conductivity than their electronic thermal conductivity. This excludes complex phononic transport mechanisms from the calculation of the figure of merit (ZT). The ZT calculations of Ca0.5L0.5MnO3 and Ca0.5L0.5−x Bi x MnO3 are corrected by the ratio of the transport relaxation time of electrical conductivity to the transport relaxation time of electronic thermal conductivity by the Lorenz number, resulting in ZT values of 0.063 and 0.327 at a temperature of 800 K, respectively.

Magneto-plasmon of AA-stacked bilayer graphene nanoribbons at finite temperature

Magneto-band structures of AA-stacked bilayer graphene nanoribbons with armchair and zigzag (BLAGNR and BLZGNR) edges are calculated by the pz-orbital tight-binding model. At zero field, BLAGNR is a semiconductor or metal determined by its width. Magnetic field can reduce band-gap and exhibit crossing bands. For metallic BLZGNRs, crossing bands shift and parabolic bands are separated more widely with magnetic field; however, partial flat bands keep unchanged. Temperature significantly increases plasmon frequency and strength for semiconducting BLAGNRs, but slightly affects those of metallic BLAGNRs and BLZGNRs. Magnetic field, for gapped BLAGNRs, can effectively reduce the threshold temperature for inducing plasmon. Whether magneto-plasmon frequency and critical momentum of dispersion relation increase or decrease is strongly sensitive to nanoribbon’s geometry. Field-modulated plasmons with a wide range of frequency could provide potential applications in waveguides and optical sensors.