Search

Insulating fillers are often mixed with polymer dielectrics in order to improve dielectric breakdown strength. However, filler-loaded dielectric films show higher variability, which limits their large-scale utility and application. In this paper, we present a computational framework for predicting the spatial distribution of electrical tree formation in a polymer dielectric in the presence of insulating fillers. The framework is based on the numerical solution of Poisson's equation inside the dielectric film and the application of the Monte Carlo model for finding the electric breakdown path. The simulation results explain several puzzles associated with the breakdown of polymer-filler dielectrics, such as (1) the improved breakdown strength with insulating filler mixing, (2) high degree of variability in breakdown measurements, and (3) effects of filler size, shape and position on the breakdown. In addition, the proposed framework not only provides the physical understanding of the breakdown process but also inspires new design concepts to improve the breakdown strength of the polymer composite.

We study how vacancies alter the effective primitive cell bands projected out of supercell eigenstates via Brillouin zone unfolding. Two types of vacant primitive cells are of particular interest: Fully vacant, in which all atoms in a single cell are missing; and net fully vacant, in which the atoms comprising a full set for a single cell are missing from more than one cell. We find that a fully vacant primitive cell and a net fully vacant primitive cell have the same effect on the primitive cell bands. We show that the probability reduction for any primitive cell band is the same, regardless of band or wavevector in the primitive cell Brillouin zone, for both fully and net fully vacant primitive cells. We illustrate these results with a two-band model.

In recent years, alloy electronic structure calculations based on supercell Brillouin zone unfolding have become popular. There are a number of formulations of the method which on the surface might appear different. Here we show that a discrete real-space description, based on discrete Fourier transforms, is fully general. Furthermore, such an approach can more easily show the effects of alloy scattering. We present such a method for treating the random alloy problem. This treatment features straightforward mathematics and a transparent physical interpretation of the calculated effective (i.e., approximate) energy bands.

Many models of condensed-matter systems have interactions with unexpected features: for example, exclusively distant-neighbor spin–orbit interactions. On first inspection these interactions seem physically questionable in view of the basis states used. However, such interactions can be physically reasonable if the model is an effective one, in which the basis states are not exactly as described, but instead include components of states removed from the problem. Mathematically, an effective model results from partitioning the Hamiltonian matrix, which can be accomplished by energy-dependent or energy-independent methods. We examine effective models of both types, with a special emphasis on energy-independent approaches. We show that an appropriate choice of basis makes the partitioning simpler and more accurate. We illustrate the method by calculating the spin–orbit splitting in graphene.

Discrete versions of continuous models are central to numerical calculations in physics and engineering. A very common problem in setting up a discrete model is how to handle derivatives. There are, for example, three common approximations for the first derivative, and each embeds different properties in the discrete model. Discretizing continuous expressions simplified using rules of calculus is especially problematic, since many different discretizations can stand for the same continuous expression depending on the stage of simplification at which the discretization is carried out. The problems are resolved by requiring that the discrete model satisfies discrete versions of the properties satisfied by the continuous original. We illustrate by using some examples from undergraduate-level one-dimensional quantum mechanics.

Due to recent improvements in computing power, non-orthogonal tight-binding models have moved beyond their traditional applications in molecular electronics to nanoelectronics. These models are appealing due to their physical chemistry content and the availability of tabulated material parameterizations. There are, however, problems with them, related to their non-orthogonality, which are more serious in nanoelectronic vs molecular applications. First, the non-orthogonal basis leads to an inherent ambiguity in the charge density. More importantly, there are problems with the position matrix in a non-orthogonal basis. The position matrix must be compatible with the underlying translationally symmetric system, which is not guaranteed if it is calculated with explicit wavefunctions. In an orthogonal basis, the only way to guarantee compatibility and gauge invariance is to use diagonal position matrices, but transforming them to a non-orthogonal basis requires major computational effort in a device consisting of 103–105 atoms. We study the charge density, position matrix, and optical absorption using a non-orthogonal two-band one-dimensional model, comparing correct and approximate calculations. We find that a typical naive calculation produces highly inaccurate results, while in contrast a first-order orthogonalized basis can represent a reasonable accuracy-efficiency trade-off.

Graphene is often surrounded by different dielectric materials when integrated into realistic devices. The absence of dangling bonds allows graphene to bond weakly via the van der Waals interaction with the adjacent material surfaces and to retain its peculiar linear band structure. In such weakly bonded systems, however, the electronic properties of graphene are affected by the dielectric screening due to the long-range Coulomb interaction with the surrounding materials. Including the surrounding materials in the first principles density functional theory (DFT) calculations is computationally very demanding due to the large supercell size required to model heterogeneous interfaces. Here, we employ a multiscale approach combining DFT and the classical image-potential model to investigate the effects of screening from the surrounding materials (hBN, SiC, SiO2, Al2O3, and HfO2) on the dielectric function and charged impurity scattering limited conductivity of graphene. In this approach, the graphene layer is modeled using DFT and the screening from the surrounding materials is incorporated by introducing an effective dielectric function. The dielectric function and conductivity of graphene calculated using the simplified two-band Dirac model are compared with DFT calculations. The two-band Dirac model is found to significantly overestimate the dielectric screening and charged impurity scattering limited conductivity of graphene. The multiscale approach presented here can also be used to study screening effects in weakly bonded heterostructures of other emerging two-dimensional materials such as metal dichalcogenides.

The phonon spectra in zinc blende InAs, GaAs and their ternary alloy nanowires (NWs) are computed using an enhanced valence force field (EVFF) model. The physical and thermal properties of these nanowires such as sound velocity, elastic constants, specific heat (C v ), phonon density of states, phonon modes, and the ballistic thermal conductance are explored. The calculated transverse and longitudinal sound velocities in these NWs are ~25% and 20% smaller compared to the bulk velocities, respectively. The C v for NWs are about twice as large as the bulk values due to higher surface to volume ratio (SVR) and strong phonon confinement in the nanostructures. The temperature dependent C v for InAs and GaAs nanowires show a cross-over at 180°K due to higher phonon density in InAs nanowires at lower temperatures. With the phonon spectra and Landauer's model the ballistic thermal conductance is reported for these III---V NWs. The results in this work demonstrate the potential to engineer the thermal behavior of III---V NWs.

Full-band nanowire simulations pose significant computational challenges. Nanowires and nanostructures in general have many interfaces, may be composed of alloys, and feature confinement on a scale of a few tens of nanometers. The empirical tight-binding approach is well-suited for modeling these devices: Its basis consists of atomic-like orbitals with limited-range interactions and reasonably-sized basis sets can accurately reproduce the bands of a wide range of semiconductors. The method easily accommodates strain and electromagnetic fields. Over the years the application of the tight-binding approach to nanodevices such as superlattices and resonant-tunneling diodes has led to the development of many useful computational techniques. Recently, its application to random-alloy nanowire calculations has led to the development of approximate bandstructure methods superior to the Virtual-Crystal Approximation for these nanostructures, and its use in nanowire transmission calculations has led to a highly efficient method for transmission calculations. I discuss tight-binding models generally and then give a more in-depth discussion of the recent developments in tight-binding models as applied to nanowires.

Atomistic disorder such as alloy disorder, surface roughness and inhomogeneous strain are known to influence electronic structure and charge transport. Scaling of device dimensions to the nanometer regime enhances the effects of disorder on device characteristics and the need for atomistic modeling arises. In this work SiGe alloy nanowires are studied from two different points of view: (1) Electronic structure where the bandstructure of a nanowire is obtained by projecting out small cell bands from a supercell eigenspectrum and (2) Transport where the transmission coefficient through the nanowire is computed using an atomistic wave function approach. The nearest neighbor sp3d5s* semi-empirical tight-binding model is employed for both electronic structure and transport. The connection between dispersions and transmission coefficients of SiGe random alloy nanowires of different sizes is highlighted. Localization is observed in thin disordered wires and a transition to bulk-like behavior is observed with increasing wire diameter.

In part I, the development and deployment of a general nanoelectronic modeling tool (NEMO 3-D) has been discussed. Based on the atomistic valence-force field and the sp3d5s* nearest neighbor tight-binding models, NEMO 3-D enables the computation of strain and electronic structure in nanostructures consisting of more than 64 and 52 million atoms, corresponding to volumes of (110 nm)3 and (101 nm)3, respectively. In this part, successful applications of NEMO 3-D are demonstrated in the atomistic calculation of single-particle electronic states of the following realistically sized nanostructures: 1) self-assembled quantum dots (QDs) including long-range strain and piezoelectricity; 2) stacked quantum dot system as used in quantum cascade lasers; 3) SiGe quantum wells (QWs) for quantum computation; and 4) SiGe nanowires. These examples demonstrate the broad NEMO 3-D capabilities and indicate the necessity of multimillion atomistic electronic structure modeling.

Often one needs to calculate the evolution time of a state under a Hamiltonian with no explicit time dependence when only numerical methods are available. In cases such as this, the usual application of Fermi's golden rule and first-order perturbation theory is inadequate as well as being computationally inconvenient. Instead, what one needs are conditions under which the evolution time may be obtained from the easily calculated energy uncertainty. This work derives some general conditions for obtaining the evolution time from the energy uncertainty.

Empirical tight-binding (ETB) methods are widely used in atomistic device simulations. Traditional ways of generating the ETB parameters rely on direct fitting to bulk experiments or theoretical electronic bands. However, ETB calculations based on existing parameters lead to unphysical results in ultrasmall structures like the As-terminated GaAs ultrathin bodies (UTBs). In this work, it is shown that more transferable ETB parameters with a short interaction range can be obtained by a process of mapping ab initio bands and wave functions to ETB models. This process enables the calibration of not only the ETB energy bands but also the ETB wave functions with corresponding ab initio calculations. Based on the mapping process, ETB models of Si and GaAs are parameterized with respect to hybrid functional calculations. Highly localized ETB basis functions are obtained. Both the ETB energy bands and wave functions with subatomic resolution of UTBs show good agreement with the corresponding hybrid functional calculations. The ETB methods can then be used to explain realistically extended devices in nonequilibrium that cannot be tackled with ab initio methods.

We present a simple circuit that performs Fermi-Dirac integration. The core of the circuit is a differential amplifier that we show is a model of a Fermi-Dirac system. The construction and characterization of the circuit requires students to apply knowledge of device physics, transistor circuit theory, and signal processing. The circuit construction is relatively simple and requires only a few components found in most electronics laboratories.

A ten-band sp 3 s ∗ second-nearest-neighbor tight-binding model has been used to model the electronic structure of various Al x Ga 1 − x As quantum cascade laser gain media. The results of the calculations have been compared with experimental emission wavelength data, and it has been shown that the model predicts the photon energies at the peaks in the gain coefficient spectra agreeing, on average, to within 4 meV of the experimental values. Comparison of the results of the calculations with results from a two-band k → ⋅ p → model shows that the tight-binding model is able to find the X -like states simultaneously with the Γ -like states. These X -like states were found to be strongly localized within the barriers. Finally, the model has also been applied to InAs/AlSb and InAs/AlSb/GaSb QCLs.

The discretized Schrodinger equation is one of the most commonly employed methods for solving one-dimensional quantum mechanics problems on the computer, yet many of its characteristics remain poorly understood. The differences with the continuous Schrodinger equation are generally viewed as shortcomings of the discrete model and are typically described in purely mathematical terms. This is unfortunate since the discretized equation is more productively viewed from the perspective of solid-state physics, which naturally links the discrete model to realistic semiconductor quantum wells and nanoelectronic devices. While the relationship between the discrete model and a one-dimensional tight-binding model has been known for some time, the fact that the discrete Schrodinger equation admits analytic solutions for quantum wells has gone unnoted. Here we present a solution to this new analytically solvable problem. We show that the differences between the discrete and continuous models are due to their fundamentally different bandstructures, and present evidence for our belief that the discrete model is the more physically reasonable one.

Atomistic computations of electronic properties for nanostructures with strain (such as self-assembled quantum dots) typically consist of two components - a calculation of the individual atomic positions and the eigenstates of interest in the resulting Hamiltonian. Such simulations ultimately require artificial boundary conditions either through a truncation of the simulation domain or by the imposition of periodic boundary conditions, which necessarily introduce inaccuracies in both components of the computation. In simulations that include up to about 20 million atoms, it is demonstrated that the simulation domain truncation has little impact on the direct computation of the electronic energies but causes considerable inaccuracies in the calculation of the atomic positions unless the simulation domain is made much larger than the central quantum dot structure. The long-range nature of the lattice distortions induced by lattice mismatch is consequently expected to significantly alter the electronic structure of nearby quantum dots.

Disorder-induced broadening of the conduction and valence band eigenenergies is calculated for an ensemble of dome-shaped InGaAs quantum dots of diameter 20nm using an sp3d5s* tight binding model.

Explicit sequences that approach the Dirac delta function and its derivatives are often helpful in presenting generalized functions. We present a method by which a finite difference formula may be easily converted into a sequence that approaches a derivative of the Dirac delta function in one dimension. In three dimensions, we employ a sequence for the Dirac delta function based on a uniformly charged sphere of infinitesimal radius and infinite charge density and show that the charge density of an electric dipole is (in the sense of a generalized function) equal to −(∂/∂z)δ3(r). We use this result to derive Gauss’ law in a dielectric medium directly from the charge densities, without using the potentials.

We discuss the two-capacitor problem found in many introductory physics texts in which there appears to be missing energy in an ideal, zero-resistance circuit, following the sudden charging of one capacitor from another. The paradox of this missing energy is traditionally ascribed to finite-resistance wires, the initial assumption of an ideal circuit and the rapid nature of the charging notwithstanding. By treating radiative effects in the simplest approximation, we show that the paradox is really nothing more than an inappropriately applied lumped-parameter model. In particular, we show that in the zero-resistance circuit, radiation fully accounts for all of the energy lost. To explore radiative effects in more realistic circuits, we also discuss numerical examples that include a small resistance and inductance.

Electronic device scaling is ultimately limited by atomic dimensions. The simulation of electronic structure and electron transport on these length scales must be fundamentally quantum mechanical. This leads to computational models that account for fundamental physical interactions using an atomistic basis and tax even the largest available supercomputer when simulating measurable devices. The prototype development of a software tool that enables this class of simulation is presented. Realistically sized structures contain one million to tens of millions of atoms that need to be represented with an appropriate basis. The resulting sparse complex Hamiltonian matrix is of the order to tens of millions. A custom matrix–vector multiplication algorithm that is coupled to a Lanczos and/or Rayleigh–Ritz eigenvalue solver has been developed and ported to a Beowulf cluster as well as an Origin 2000. First benchmarking results of these algorithms as well as the first results of quantum dot simulations are reported.

The broadening of the conduction and valence band edges due to compositional disorder in alloyed materials of finite extent is studied using an sp3s* tight binding model. Two sources of broadening due to configuration and concentration disorder are identified. The concentrational disorder dominates for systems up to at least one million atoms and depends on problem size through an inverse square root law. Significant differences (over 12 meV) in band edge energies are seen depending on choice of granularity of alloy clusters.

Surface passivation is of great technological importance due to the increasing miniaturisation of electronic devices. It has been known for many years that under certain conditions surface states can form; when they do so in a quantum well (QW) the result is an unbound (i.e., evanescent) state in the QW. Such surface states are generally undesirable, so a good physical understanding of them is important. A simple single-p-orbital valence band model is used with two types of surface passivation to examine surface states in a QW: (1) an energy upshift added to the terminal atoms; and (2) explicit passivation by an s-orbital on each end of the QW. These models show these unbound/evanescent QW states can occur in both models; that in them the wavefunction is bound to the terminal atoms; and that the existence of these states is connected to the effective valence-band offset between the terminal atoms and the bulk QW.

Thermal properties are of great interest in modern electronic devices and nanostructures. Calculating these properties is straightforward when the device is made from a pure material, but problems arise when alloys are used. Specifically, only approximate bandstructures can be computed for random alloys and most often the Virtual Crystal Approximation (VCA) is used. Unfolding methods [T. B. Boykin, N. Kharche, G. Klimeck, and M. Korkusinski, J. Phys.: Condens. Matt. 19, 036203 (2007).] have proven very useful for tight-binding calculations of alloy electronic structure without the problems in the VCA, and the mathematical analogy between tight-binding and valence-force-field approaches to the phonon problem suggest they be employed here as well. However, there are some differences in the physics of the two problems requiring modifications to the electronic structure approach. We therefore derive a phonon alloy bandstructure (vibrational mode) approach based on our tight-binding electronic structure method, modifying the band-determination method to accommodate the different physical situation. Using the method, we study In$_x$Ga$_{1-x}$As alloys and find very good agreement with available experiments., Main paper with attached supplemental material

The graphene nanoribbon (GNR) tunneling field effect transistor (TFET) has been a promising candidate for a future low power logic device due to its sub-60 mV/dec subthreshold characteristic and its superior gate control on the channel electrons due to its one-dimensional nature. Even though many theoretical studies have been carried out, it is not clear that GNR TFETs would outperform conventional silicon metal oxide semiconductor field effect transistors (MOSFETs). With rigorous atomistic simulations using the p/d orbital tight-binding model, this study focuses on the optimization of GNR TFETs by tuning the doping density and the size of GNRs. It is found that the optimized GNR TFET can operate at a half of the supply voltage of silicon nanowire MOSFETs in the ballistic limit. However, a study on the effects of edge roughness on the performance of the optimized GNR TFET structure reveals that experimentally feasible edge roughness can deteriorates the on-current performance if the off-current is normalized with the low power requirement specified in the international technology roadmap for semiconductors. (C) 2014 AIP Publishing LLC.

In quantum mechanics texts one sometimes encounters the unqualified (and generally untrue) assertion that the solution of the Schrodinger equation for a charged particle in the presence of an externally applied vector potential can be found from that in the absence of the vector potential simply via multiplication with an r-dependent phase factor. The confusion caused by this assertion is only increased when one examines the expression for the matrix elements of the Hamiltonian including electromagnetic interactions (via the vector potential) appropriate for a tight-binding model, for this expression indeed takes a different form from that of the usual minimal coupling Hamiltonian in continuous space. Motivated by the tight-binding (i.e., discretized) result, we derive its continuous-space analog. We show that the aforementioned perplexing assertion actually arises from a confusion between the wavefunction and the matrix elements of the continuous-space Hamiltonian in the position basis and discuss both the tight-binding and continuous-space expressions.

Heterostructures such as resonant tunneling diodes, quantum well photodetectors and lasers, and cascade lasers break the symmetry of the crystalline lattice. Such break in lattice symmetry causes a strong interaction of heavy-, light- and split-off hole bands. A resonant tunneling diode is used as a vehicle to study hole transport in heterostructures including the subband dispersion transverse to the main transport direction. Four key findings are demonstrated: (1) the heavy and light hole interaction is shown to be strong enough to result in dominant current flow off the Γ zone center (more holes flow through the structure at an angle than straight through), (2) explicit inclusion of the transverse momentum in the current integration is needed, (3) most of the current flow is due to injection from heavy holes in the emitter, and (4) the dependence on the angle φ of the transverse momentum k is weak. Two bandstructure models are utilized to demonstrate the underlying physics: (1) independent/uncoupled heavy-, light- and split-off bands, and (2) second-nearest neighbor sp3s* tight-binding model. Current–voltage (I–V) simulations including explicit integration of the total energy E, transverse momentum | k | and transverse momentum angle φ are analyzed. An analytic formula for the current densityJ (k) as a function of transverse momentum k is derived and utilized to explain the three independent mechanisms that generate off-zone-center current flow: (1) nonmonotonic (electron-like) hole dispersion, (2) different quantum well and emitter effective masses, and (3) momentum-dependent quantum well coupling strength. The analytic expression is also used to generate a complete I–V characteristic that compares well to the full numerical solution. The Fermi level and temperature dependence on the I–V is examined. Finally a simulation is compared to experimental data.

We show how the exact representation of the operator $\mathrm{exp}(i\mathbf{q}\mathbf{\ensuremath{\cdot}}\mathbf{r})$ in the Bloch sum basis used in the empirical tight-binding method follows from the transformation law obeyed by cell-periodic operators, such as the Hamiltonian. From this representation, we derive matrix elements of its product with cell-periodic operators, since product operators of this type arise when the crystal is subject to various perturbations. Using these results, we calculate the expression for the linear transverse dielectric tensor.

The conventional derivation of the self-inductance is often unnecessarily difficult and abstract. This is especially true of inductance calculations for circuits in which the current is distributed (composite circuits), rather than filamentary. The traditional approach for these circuits is based upon the concept of partial flux linkages, and hence tends to be very confusing even in the simplest of cases. Here we present a more straightforward treatment, emphasizing the equivalent-circuit nature of the self-inductance, yet still closely tied to Faraday’s law. We demonstrate the transparency of our approach both by contrasting it with the usual calculation of the inductance of a solid-core coaxial cable and by applying it to the more complicated case of a coaxial cable with inner and outer conductors of finite thickness.

Simple heterostructure models are often employed due to their computational efficiency. The approximations involved are not, however, well understood; this is particularly true with respect to the interfaces. In order to clarify this situation we study two different monoatomic (single-band) models and two different diatomic (two-band) models all having identical dispersion relations. We study in detail the relationships between the one- and two-band models, showing that their points of agreement and disagreement arise directly from the handling of the interfaces.

We derive and study exact analytic expressions for the effective masses of the conduction--and all three hole--bands at \ensuremath{\Gamma} in the spin-orbit, second-nearest-neighbor ${\mathrm{sp}}^{3}{s}^{*}$ model. Using these expressions we determine parameters for six common III-V materials (GaAs, AlAs, GaSb, AlSb, InAs, and InP), tailored for [001]-oriented heterostructure calculations. Beyond their use in fitting band structures, the effective-mass formulas show that the second-nearest-neighbor ${\mathrm{sp}}^{3}{s}^{*}$ model is not without limitations. We show that there is an upper bound on the reproducible electron--light-hole effective-mass mismatch, so that even this model may not be sufficient for certain materials.

Combining molecular dynamics and quantum transport simulations, we study the degradation of mobility in graphene nanoribbons caused by substrate-induced ripples. First, the atom coordinates of large-scale structures are relaxed such that surface properties are consistent with those of graphene on a substrate. Then, the electron current and low-field mobility of the resulting non-flat nanoribbons are calculated within the Non-equilibrium Green's Function formalism in the coherent transport limit. An accurate tight-binding basis coupling the sigma- and pi-bands of graphene is used for this purpose. It is found that the presence of ripples decreases the mobility of graphene nanoribbons on SiO2 below 3000 cm(2)/Vs, which is comparable to experimentally reported values. (C) 2013 AIP Publishing LLC.

Semi-Empirical Tight Binding (TB) is known to be a scalable and accurate atomistic representation for electron transport for realistically extended nano-scaled semiconductor devices that might contain millions of atoms. In this paper an environment-aware and transferable TB model suitable for electronic structure and transport simulations in technologically relevant metals, metallic alloys, metal nanostructures and metallic interface systems is described. Part I of this paper describes the development and validation of the new TB model. The new model incorporates intra-atomic diagonal and off-diagonal elements for implicit self-consistency and greater transferability across bonding environments. The dependence of the on-site energies on strain has been obtained by appealing to the Moments Theorem that links closed electron paths in the system to energy moments of angular momentum resolved local density of states obtained ab-initio. The model matches self-consistent DFT electronic structure results for bulk FCC metals with and without strain, metallic alloys, metallic interfaces and metallic nanostructures with high accuracy and can be used in predictive electronic structure and transport problems in metallic systems at realistically extended length scales, Comment: 34 pages including appendices, parameters and supplemental material in double spaced format

The k\ensuremath{\cdot}p perturbation expansion for the energy of a nondegenerate band, familiar from solid-state texts, provides a convenient method for calculating the effective mass of the band in terms of the eigenstates of the system. As such, it would seem particularly useful in fitting the parameters of an empirical tight-binding model. Unfortunately, this expression is incorrect when applied to a model having an incomplete basis. We find that the correct expression may be derived from the usual result if the incompleteness of the basis is properly taken into account. In addition, since degenerate bands are often of interest, we discuss the correct calculation of their curvatures.

Resonant‐tunneling diodes designed to have features in their current‐density‐voltage (J‐V) characteristics in addition to the main peak often incorporate triple‐barrier structures. In designing such structures, much attention is paid to the alignment of the quasibound levels in the two wells in order to achieve additional peaks or kinks in the J‐V curve. Unfortunately, many such devices fail to display these additional features. It is commonly thought that this failure is solely due to the limitations of coherent tunneling models, but this is not always the case. Here we demonstrate that the simplest and most commonly employed approximation for the tunneling current density (the one‐dimensional approximation) is often incorrect for triple‐barrier devices and that when a more accurate approximation (the two‐dimensional approximation) is used the J‐V characteristics can be markedly different.

We present here what is, to our knowledge, a completely new and general solution expression for the complementary solution of an arbitrary Nth order linear homogeneous constant-coefficient difference equation which, unlike the solution expressions usually presented in textbooks, does not a priori assert the specific structural form of the solution. This method easily handles the case of repeated zero roots, a case of practical importance for which the classical solution expression fails, as recently shown by Johnson. Furthermore, we show that both the classical solution expression, and Johnson's “singular solution” expression for the case of repeated zero roots, are special cases of our more general expression. Finally, we present an example illustrating the interrelationships amongst the different solution expressions as well as the solution obtained via the generating-function method.

One expects that the one-dimensional approximation for the resonant-tunneling-diode tunneling current should be very accurate in the InAs/AlSb materials system, with its large barriers and largely \ensuremath{\Gamma}-like tunneling. We study this approximation as well as the two-dimensional expression, which takes into account the explicit dependence of the transmission coefficient on the magnitude of the in-plane wave vector ${\mathbf{k}}_{\mathrm{\ensuremath{\parallel}}}$. We find that even here the one-dimensional approximation fails, producing curves that are qualitatively very different from those of the two-dimensional approximation and study the reasons for the differences. We also briefly examine the angular dependence of the transmission coefficients, the results indicating that the two-dimensional approximation is likely to be fairly good for the structures studied.

The single π−orbital model for graphene has been successful for extended, perfectly flat sheets. However, it cannot model hydrogen passivation, multi-layer structures, or rippled sheets. We address these shortcomings by adding a full complement of d-orbitals to the traditional {s,p} set. To model strain behavior and multi-layer structures we fit scaling exponents and introduce a long-range scaling modulation function. We apply the model to rippled graphene nanoribbons and bi-layer graphene sheets.