Airy Function Approach Research Articles

Nonlinear optical susceptibilities in InxGa1−xN/GaN hexagonal single quantum well under applied electric field

The hexagonal single quantum well fabricated using the nitride semiconductor heterostructures GaN/InxGa1−xN/GaN contains a built-in electric field as a result of lattice strain in this wurtzite structure. The eigenstates of the quantum well are obtained by solving the effective mass equation under applied electric field along the growth direction and opposite to it by the analytic Airy function approach. Analytic expressions for the nonlinear optical properties up to third-order are obtained by using the compact density matrix approach taking a quantum well containing two levels. The Stark shifted envelope functions yield large dipole matrix elements which enhance the optical susceptibilities. The internal and external electric fields significantly blue shift the susceptibilities. The susceptibilities are increased under electric field along the growth direction, while it decrease under the opposite direction.

Transmittance and Tunneling Current through a Trapezoidal Barrier under Spin Polarization Consideration

The transmittance and tunneling current in heterostructures under spin polarization consideration were studied by employing a zinc-blended structure for the heterostructures. An electron tunnels through a potential barrier by applying a bias voltage to the barrier, which is called the trapezoidal potential barrier. In order to study the transmittance, an Airy wave function approach was employed to find the transmittance. The obtained transmittance was then utilized to compute the tunneling current by using a Gauss quadrature method. It was shown that the transmittances were asymmetric with the incident angle of the electron. It was also shown that the tunneling currents increased as the bias voltage increased.

Plane strain cylindrical indentation of functionally graded half-plane with exponentially varying shear modulus in the presence of residual surface tension

A functionally graded half-plane with shear modulus varying exponentially along the direction normal to the surface and surface effects accounted through Gurtin–Murdoch model, indented by long rigid smooth cylindrical indenter is solved to understand the effect of material inhomogeneity and surface effects on indentation response. The Green’s function relating surface load to surface displacement under plane strain condition is obtained semi-analytically through the combination of Airy stress function approach and Fourier transforms and utilized to solve the contact problem. The solution is used to study the effect of inhomogeneity through grading parameter and surface effects through residual surface tension based intrinsic length scale on the contact pressure, contact size and in-plane normal stress on the surface responsible for cracks during indentation.

Comparison of Electron Transmittance and Tunneling Current through a Trapezoidal Potential Barrier with Spin Polarization Consideration by using Analytical and Numerical Approaches

In this study, we report an analytical calculation of electron transmittance and polarized tunneling current in a single barrier heterostructure of a metal-GaSb-metal by considering the Dresselhaus spin orbit effect. Exponential function, WKB method and Airy function were used in calculating the electron transmittance and tunneling current. A Transfer Matrix Method, as a numerical method, was utilized as the benchmark to evaluate the analytical calculation. It was found that the transmittances calculated under exponential function and Airy function is the same as that calculated under TMM method at low electron energy. However, at high electron energy only the transmittance calculated under Airy function approach is the same as that calculated under TMM method. It was also shown that the transmittances both of spin-up and spin-down conditions increase as the electron energy increases for low energies. Furthermore, the tunneling current decreases with increasing the barrier width.

Airy function approach and Numerov method to study the anharmonic oscillator potentials V(x) = Ax2α + Bx2

The numerical solutions of the time independent Schrödinger equation of different one-dimensional potentials forms are sometime achieved by the asymptotic iteration method. Its importance appears, for example, on its efficiency to describe vibrational system in quantum mechanics. In this paper, the Airy function approach and the Numerov method have been used and presented to study the oscillator anharmonic potential V(x) = Ax2α + Bx2, (A>0, B<0), with (α = 2) for quadratic, (α =3) for sextic and (α =4) for octic anharmonic oscillators. The Airy function approach is based on the replacement of the real potential V(x) by a piecewise-linear potential v(x), while, the Numerov method is based on the discretization of the wave function on the x-axis. The first energies levels have been calculated and the wave functions for the sextic system have been evaluated. These specific values are unlimited by the magnitude of A, B and α. It’s found that the obtained results are in good agreement with the previous results obtained by the asymptotic iteration method for α =3.

Effect of Thickness and Temperature of SiO2 Layer on Leakage Currents in MOS Capacitor Materials with High Dielectric Constant by Involving the Charge Trap

Modeling of the leakage current in a field-effect transistor metal-oxide-semiconductor (MOSFET) with high dielectric material has been developed by taking into account the effect of charge traps formed at the interface of high-k material/SiO2. Transmittance calculated using Airy wave function approach and involving the anisotropic electron mass and the effect of coupling between transverse and longitudinal energy represented by the speed of the electrons in the metal gate. Transmittance obtained is then used to calculate the leakage current in the structure of n+Poly-Si/HfSiOxN/trap/SiO2/p-Si for oxide voltage variations, temperature, and thickness of the SiO2. From the calculation that the leakage current decreases with decreasing of oxide voltage and increasing of the thickness of the oxide layer of SiO2. Also obtained that the temperature of the device does not give a great influence on the change of leakage current. Keywords: Leakage current, electron speed, charge traps

Static Deformation due to a Long Tensile Fault of Finite Width in an Isotropic Half-Space Welded with an Orthotropic Half-Space

Closed-form analytical expressions for displacements and stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space caused by a tensile fault of finite width located at an arbitrary distance from the interface in the isotropic half-space are obtained. The Airy stress function approach is used to obtain the expressions for the stresses and displacements. The vertical tensile fault is considered graphically. The variations of the displacements with the distance from the fault and with depth for various cases have been studied graphically. Also horizontal and vertical displacement of the surface are presented graphically.

Modified Airy function method modelling of tunnelling current for Schottky barrier diodes on silicon carbide

We present a simple method for analysing the tunnelling current through Schottky barrier diodes on SiC, based on the modified Airy function (MAF) approach. The MAF method is accurate for linear-shaped barriers which is the case for the top of the Schottky barrier diodes. The results have been compared with those obtained by the conventional Wentzel–Kramers–Brillouin (WKB). This study proves that the WKB method is valid for the Schottky barrier diodes with and without the incorporation of Schottky barrier lowering under low or high bias voltage.

Transmission coefficients for the exact triangular barrier: an exact general analytical theory that can replace Fowler & Nordheim's 1928 theory

In field electron emission theory, evaluating the transmission coefficient D ET for an exact triangular (ET) potential energy barrier is a paradigm problem. This paper derives a compact, exact, general analytical expression for D ET , by means of an Airy function approach that uses a reflected barrier and puts the origin of coordinates at the electron's outer classical turning point. This approach has simpler mathematics than previous treatments. The expression derived applies to both tunnelling and ‘flyover’ (wave-mechanical transmission over the barrier), and is easily evaluated by computer algebra. The outcome is a unified theory of transmission across the ET barrier. In different ranges of relevant physical parameters, the expression yields different approximate formulae. For some ranges, no simple physical dependences exist. Ranges of validity for the most relevant formulae (including the Fowler–Nordheim 1928 formula for D ET ) are explored, and a regime diagram constructed. Previous treatments are assessed and some discrepancies noted. Further approximations involved in deriving the Fowler–Nordheim 1928 equation for current density are stated. To assist testing of numerical procedures, benchmark values of D ET are stated to six significant figures. This work may be helpful background for research into transmission across barriers for which no exact analytical theory yet exists.

Scratch test model for the determination of fracture toughness

Abstract We revisit the scratch test within the framework of linear elastic fracture mechanics. In the analysis, we employ an Airy stress function approach to determine stresses and displacement in the vicinity of the scratch-blade–material interface, which serve as input for the evaluation of the energy release rate by means of the J-Integral. In contrast to previous models, the energy release rate thus found scales with the sum of the applied forces squared. This entails a linear relation between the applied forces and w d , where w is the scratch width and d the scratch depth. This analytical scaling is validated using experimental scratch data on cement paste and sandstone, which shows that the proposed approach provides a convenient way to determine the fracture toughness from scratch tests carried out with different scratch widths and depths.

Static deformation due to a long buried dip-slip fault in an isotropic half-space welded with an orthotropic half-space

Closed-form analytical expressions for the displacements and the stresses at any point of a two-phase medium consisting of a homogeneous, isotropic, perfectly elastic half-space in welded contact with a homogeneous, orthotropic, perfectly elastic half-space due to a dip-slip fault of finite width located at an arbitrary distance from the interface in the isotropic half-space are obtained. The Airy stress function approach is used to obtain the expressions for the stresses and the displacements. The case of a vertical dip-slip fault is considered in detail. The variations of the displacements with the distance from the fault and with depth have been shown graphically.

Material tailoring and analysis of functionally graded isotropic and incompressible linear elastic hollow cylinders

We use the Airy stress function to derive exact solutions for plane strain deformations of a functionally graded (FG) hollow cylinder with the inner and the outer surfaces subjected to different boundary conditions, and the cylinder composed of an isotropic and incompressible linear elastic material. For the shear modulus given by either a power law or an exponential function of the radius r, we derive explicit expressions for stresses, the hydrostatic pressure and displacements. Conversely, we find the variation with r of the shear modulus for a linear combination of the radial and the hoop stresses to have a preassigned variation in the cylinder; this inverse problem is usually called material tailoring. The shear modulus found while solving the inverse problem must be positive everywhere. Results for a few problems are computed and presented graphically. It seems that the Airy stress function approach is used here for the first time to analyze two-dimensional problems for incompressible materials. When studying axisymmetric deformations of an FG cylinder, it is found that for the hoop stress to be uniform through the cylinder thickness the shear modulus must be proportional to the radial coordinate r as found earlier by Batra [Batra RC. Optimal design of functionally graded incompressible linear elastic cylinders and spheres. AIAAJ 2008;46(8):2005–7.] and for the maximum in-plane shear stress to be constant the shear modulus must vary as r 2 . The expression for the maximum in-plane shear stress in terms of pressures and the radii of the inner and the outer surfaces of the cylinder is a universal result valid for all materials for which the shear modulus is proportional to r 2

Current density in generalized Fibonacci superlattices under a uniform electricfield

We present an exhaustive study on tunneling and electrical conduction in an electrically biasedGaAs–AlyGa1−yAs generalized Fibonacci superlattice. The study is based on transfer matrix formalism usingan Airy function approach and provides an exact calculation of the current density in thecase of quasi-periodic multibarrier systems. The results suggest the use of suchquasi-periodic systems in perfect band-pass or band-eliminator (of extremely low width)circuitry. We have clearly demonstrated the resonance-type peaks and negativedifferential conductivity regimes in such systems. It has also been found thatquasi-periodicity favors sharp negative differential conductivity peaks compared to those inperiodic superlattices and thus have profound importance in device applications.

A theoretical study of resonant tunneling characteristics in triangular double-barrier diodes

Resonant tunneling characteristics of triangular double-barrier diodes have been investigated systematically in this Letter, using Airy function approach to solve time-independent Schrödinger function in triangular double-barrier structures. Originally, the exact analytic expressions of quasi-bound levels and quasi-level lifetime in symmetrical triangular double-barrier structures have been derived within the effective-mass approximation as a function of structure parameters including well width, slope width and barrier height. Based on our derived analytic expressions, numerical results show that quasi-bound levels and quasi-level lifetime vary nearly linearly with the structure parameters except that the second quasi-level lifetime changes parabolically with slope width. Furthermore, according to our improved transmission coefficient of triangular double-barrier structures under external electric field, the current densities of triangular double-barrier diodes with different slope width at 0 K have been calculated numerically. The results show that the N-shaped negative differential resistance behaviors have been observed in current–voltage characteristics and current–voltage characteristics depend on the slope width.

Generalized electron emission model for field, thermal, and photoemission

Analysis of electron emission from photocathodes, field emitters under extreme fields, or thermionic emitters operated at reduced temperature, utilize thermionic (Richardson) or field emission (Fowler–Nordheim) approximations which become inaccurate for such atypical conditions. The computational overhead of advanced numerical transport models make them ill-suited for data analysis or simulations of extended areas of photocathode and thermionic emitters, or for nonplanar field emitters. In this letter, an analytic thermal-field emission equation is given for which the Fowler–Nordheim and Richardson–Laue–Dushman equations are asymptotic limits. The methodology can analytically address “warm” field and “cool” thermionic emission, photoemission, and electron transport between interfaces (e.g., Schottky barriers). The approximations developed are compared to an exact evaluation (the modified airy function approach).

Photon assisted field emission from a silicon emitter

A novel method for the high-frequency modulation of a semiconductor field emitter array (FEA), as needed by compact high power microwave and millimeter wave tubes, is qualitatively analyzed. The model examines a FEA held at the threshold of emission by an applied gate potential from which current pulses are triggered by the application of a laser pulse on the backside of the semiconductor. Such an arrangement produces electron bunches (“density modulation”) at GHz frequencies without suffering from the restriction in reduced emission area and small unit cell geometry imposed by the high capacitance of field emitters. The analysis proceeds by first developing an analytical model of the emission from a silicon tip using a modified WKB approach to the tunneling current, which is validated by the more exact Airy function approach to solving Schrödinger’s equation. The effects of band bending are explicitly accounted for. The resulting relations are used to estimate emission from a single hyperbolic structure, and generalized to an array where a distribution in tip radii and work function is possible. Using a simple relationship between the incident photon flux and the resultant electron density at the emission site, an estimation of the tunneling current is made. Finally, a brief description of the operation and design of such a “photon-assisted field emission device” is given.

Electron Transmission Through Modified Schottky Barriers

AbstractThe effects of a Coulomb-like potential in the Schottky barrier existing between a material-diamond interface is analyzed. The inclusion is intended to mimic the effects of an ionized trap within the barrier, and therefore to account for charge injection into the conduction band of diamond via a Poole-Frenkel transport mechanism. The present treatment is to provide a qualitative account of the increase in current density near the inclusion, which can be substantial. The model is first reduced to an analytically tractable one-dimensional tunneling problem addressable by an Airy Function approach in order to investigate the nature of the effect. A more comprehensive numerical approach is then applied. Finally, statistical arguments are used to estimate emission site densities using the results of the aforementioned analysis.

Analysis of a photon assisted field emission device

A field emitter array held at the threshold of emission by a dc gate potential from which current pulses are triggered by the application of a laser pulse on the backside of the semiconductor may produce electron bunches (“density modulation”) at gigahertz frequencies. We develop an analytical model of such optically controlled emission from a silicon tip using a modified Wentzel–Kramers–Brillouin and Airy function approach to solving Schrödinger’s equation. Band bending and an approximation to the exchange-correlation effects on the image charge potential are included for an array of hyperbolic emitters with a distribution in tip radii and work function. For a simple relationship between the incident photon flux and the resultant electron density at the emission site, an estimation of the tunneling current is made. An example of the operation and design of such a photon-assisted field emission device is given.

Simulation of Schottky barrier MOSFETs with a coupled quantum injection/Monte Carlo technique

A full-band Monte Carlo device simulator has been used to analyze the performance of sub-0.1 /spl mu/m Schottky barrier MOSFETs. In these devices, the source and drain contacts are realized with metal silicide, and the injection of carriers is controlled by gate voltage modulation of tunneling through the source barrier. A simple model treating the silicide regions as metals, coupled with an Airy function approach for tunneling through the barrier, provides injecting boundary conditions for the Monte Carlo procedure. Simulations were carried out considering a p-channel device with 270 /spl Aring/ gate length for which measurements are available. Our results show that in these structures there is not a strong interaction with the oxide interface as in conventional MOS devices and carriers are injected at fairly wide angles from the source into the bulk of the device. The Monte Carlo simulations not only give good agreement with current-voltage (I-V) curves, but also easily reproduce the subthreshold behavior since all the computational power is devoted to simulation of channel particles. The simulations also clarify why these structures exhibit a large amount of leakage in subthreshold regime, due to both thermionic and tunneling emission. Computational experiments suggest ways to modify the doping profile to reduce to some extent the leakage.

Simulation of the Influence of Interface Charge on Electron Emission

ABSTRACTSeveral materials are promising candidates for electron sources. For diamond, a tunneling interface at the back contact limits injecting charge into the conduction band, but a purely geometric model of internal field emission is inadequate to explain experimental data. The presence of a defect, modeled by a coulomb charge, within the tunneling barrier region significantly enhances transmission and, in concert with a geometrical model, may better account for observed current levels. Charge has been suggested to play a similar role in the SiO2 covering on a single tip silicon field emitter to explain experimental data. The tunneling theory in both cases is similar. In the present work, a general method for estimating electron transport and energy distributions through potential profiles, which describe both semiconductor interfaces and field emission potential barriers when a charged particle modifies the tunneling barrier, is developed. While the model is intended for treating a metal-semiconductor interface, it is cast here in terms of in a thin SiO2 coating over a silicon field emitter tip to enable qualitative comparisons with experimental data. Tunneling probabilities are found by numerically solving Schrödinger's Equation for a piece-wise linear potential using an Airy Function approach. A qualitative comparison to experimental energy distribution findings is possible by utilizing an analytical model of the field emitter tip from which current-voltage relations may be found.