Macroscopic Continuum Model Research Articles

Effects of interface phonon scattering in three-interface heterostructures

A detailed study of the electron–optical–phonon interaction in an asymmetric one-well/one-barrier heterostructure is given. Dispersion relations and phonon potential distributions for interface phonon modes are derived in the framework of the macroscopic dielectric continuum model. It is found that for intrawell relaxation processes the sum of the scattering rates by all interface polar-optical phonon modes is approximately independent of the width of the barrier layer. Consequently, a simplified Hamiltonian for electron–phonon interaction in a single quantum well can be used for scattering rate calculation in multiple heterointerface structures. The combined scattering rates by interface and confined phonon modes are compared with the results obtained in an idealized model using the bulklike phonon spectrum. The practical invalidity of the latter approximation is shown for electron kinetic energies comparable with the typical energy of optical phonons in the heterostructure.

Lntersubband Relaxation in Step Quantum Well Structures

Intersubband relaxation due to electron interactions with the localized phonon modes plays an important role for population inversion in quantum well laser structures designed for intersubband lasers operating at mid-infrared to submillimeter wavelengths. In this work, intersubband relaxation rates between subbands in step quantum well structures are evaluated numerically using Fermi's golden rule, in which the localized phonon modes including the asymmetric interface modes, symmetric interface modes, and confined phonon modes and the electron – phonon interaction Hamiltonians are derived based on the macroscopic dielectric continuum model, whereas the electron wave functions are obtained by solving the Schrödinger equation for the heterostructures under investigation. The sum rule for the relationship between the form factors of the various localized phonon modes and the bulk phonon modes is examined and verified for these structures. The intersubband relaxation rates due to electron scattering by the asymmetric interface phonons, symmetric interface phonons, and confined phonons are calculated and compared with the relaxation rates calculated using the bulk phonon modes and the Fröhlich interaction Hamiltonian for step quantum well structures with subband separations of 36 meV and 50meV, corresponding to the bulk longitudinal optical phonon energy and interface phonon energy, respectively. Our results show that for preferential electron relaxation in intersubband laser structures, the effects of the localized phonon modes, especially the interface phonon modes, must be included for optimal design of these structures.

A real-time traffic simulation system

This article studies the usefulness of parallel processing in real-time traffic-flow simulation based on continuum modeling of traffic dynamics. Computational fluid dynamics (CFDs) methods for solving simple macroscopic traffic-flow continuum models have been studied and efficiently implemented in traffic simulation codes (on serial computers) in the past. We designed a traffic-flow simulation code and mapped it onto a parallel computer architecture. This traffic simulation system is capable of simulating freeway traffic flow in real time. Tests with real traffic data collected from the freeway network in the metropolitan area of Minneapolis, MN, were used to validate the accuracy and computational rate of the parallel simulation system. The execution time for a 2-h traffic-flow simulation of about 200 619 vehicles in an 18-mi freeway, which takes 2.35 min of computer time (on a single-processor computer simulator), took only 5.25 s on the parallel traffic simulation system. This parallel system has a lot of potential for real-time traffic engineering applications.

Transfer matrix method for interface optical-phonon modes in multiple-interface heterostructure systems

Interactions of carriers with interface optical phonons dominate over other carrier–phonon scatterings in narrow quantum-well structures. Herein, a transfer matrix method is used to establish a formalism for determining the dispersion relations, electrostatic potentials, and Fröhlich interaction Hamiltonians of the interface optical phonons for multiple-interface heterostructure systems within the framework of the macroscopic dielectric continuum model. This method facilitates systematic calculations for complex structures where the conventional method is very difficult to implement. Several specific cases are treated to illustrate the advantages of the general formalism.

Planar vibrational modes in semiconductors: A simple model

A macroscopic continuum model for optical phonon modes is applied to obtain the planar vibrational modes of a monoatomic layer embedded in a homogeneous material. The differences in optical force constants and reduced masses across the atomic plane are considered within the theory as a fundamental factor to describe the modes localized at the interface. We have derived analytical expressions for the phonon spectrum, as well as for the vector displacement and electric potential as a function of coordinates. An analysis of the infrared and Raman activity for the obtained modes is given together with a comparison with recent experimental data.

Electron–optical-phonon scattering in wurtzite crystals

We present Fr\ohlich-like electron--optical-phonon interaction Hamiltonian for wurtzite crystals in the long-wavelength limit based on the macroscopic dielectric continuum model and the uniaxial model. In general, the optical-phonon branches support mixed longitudinal and transverse modes due to the anisotropy. We calculate electron scattering rate due to the optical phonons in the bulk wurtzite GaN and demonstrate that the scattering rate due to transverselike phonon processes can be strongly enhanced over a range of angles with respect to the $c$ axis. For the case of longitudinal-like modes, the anisotropy is small and the result is almost the same as that obtained with the cubic Fr\ohlich Hamiltonian.

Recent Developments in Modeling of Liquid Phase Electroepitaxy: A Continuum Approach

Liquid phase electroepitaxy (LPEE) is a low temperature, solution growth technique which has been proven to be successful in growing high quality, thick compound and alloy semiconducting single crystal layers. The availability of such high quality alloy layers would open new horizons in the production of optoelectronic and high-speed devices. Due to the technological importance of this crystal growth technique, some theoretical models for its growth process have been developed recently to provide needed information to experimentalists to develop growth configurations for reproducible desired single crystals. These models have shed light on various aspects of the LPEE growth process. This review article, with 71 references, provides the reader with an overview of recently developed macroscopic continuum models for the LPEE growth process of binary and ternary semiconductors. Fundamental equations of these models are obtained from the basic principles of electrodynamics and thermomechanics of the continuum. The models include various thermoelectric effects observed in LPEE and also incorporates microscopic surface phenomena such as surface kinetics. Results of numerical simulations are presented, and compared with available experimental data. The significance of research results are discussed.

Electron interaction with confined acoustic phonons in cylindrical quantum wires via deformation potential

The effects of phonon confinement on electron–acoustic-phonon scattering is studied in cylindrical semiconductor quantum wires. In the macroscopic elastic continuum model, the confined-phonon dispersion relations are obtained for several crystallographic directions with the two cardinal boundary conditions: free-surface and clamped-surface boundary conditions. The scattering rates due to the deformation potential interaction are obtained for these confined phonons and are compared with those of bulk-like phonons for a number of quantum wire materials. The results show that the inclusion of acoustic phonon confinement effects may be crucial for calculating accurate low-energy electron scattering rates in nanostructures. It is also demonstrated that the scattering rates may be significantly influenced by the direction of phonon propagation, especially for low-energy electrons. Furthermore, it has been found that there is a scaling rule governing the directional dependence of the scattering rates: the directions characterized by small Poisson ratios exhibit large scattering rates.

Screening effects on the confined and interface polarons in cylindrical quantum wires.

We study the contribution of confined and interface phonons to the polaron energy in quantum-well wires. We use a dispersionless, macroscopic continuum model to describe the phonon confinement in quantum wires of circular cross section. Surface phonon modes of a free-standing wire and interface phonon modes of a wire embedded in a dielectric material are also considered. Polaron energy is calculated by variationally incorporating the dynamic screening effects. We find that the confined and interface phonon contribution to the polaron energy is comparable to that of bulk phonons in the density range N=${10}^{5}$--${10}^{7}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}1}$. Screening effects within the random-phase approximation significantly reduce the electron-confined phonon interaction, whereas the exchange-correlation contribution tends to oppose this trend at lower densities. \textcopyright{} 1996 The American Physical Society.

PHONON-LIMITED MOBILITY IN A FREE-STANDING POLAR SEMICONDUCTOR QUANTUM WIRE

A macroscopic continuum model is used to study the longitudinal optical phonons in a free-standing polar quantum wire with a rectangular cross section. The Fröhlich Hamiltonian describing the electron-phonon interaction is then obtained and used to calculate the electron mobility governed by confined LO phonons by means of the memory function approach, neglecting electron-electron interaction. Numerical calculations are given for a GaAs quantum wire, showing that the mobility is enhanced at least by one order of magnitude in comparison with the bulk-LO-phonon-limited one. It is shown that the contribution of intersubband scattering is important in wires of large width at high temperature.

A continuum model for liquid phase electroepitaxy

This paper presents a macroscopic continuum model for liquid phase electroepitaxial growth of single crystal semiconductors. The governing equations and associated boundary and interface conditions of the model are obtained from the fundamental principles of electrodynamics and thermomechanics of continua. The constitutive equations of the substrate/source and the liquid phase are derived from an irreversible rational thermodynamic theory. By means of systematic simplifications, special forms of the governing equations and associated interface conditions are presented in order to obtain tractable equations and gain physical insight for various thermoelectric effects involved in the process. The formulation presented here is valid for general LPEE growth processes with any configuration, and takes into account electromigration and the well-known thermoelectric effects such as Joule, Peltier, Thomson, Dufour, and Sorel. The fundamental equations derived here can also be used to model the growth process of ternary compound semiconductors, either directly or with modifications depending on the type of compositions considered.

Phonon confinement and screening effects on the polaron energy in quantum wires

We study the contribution of confined phonons to the polaron energy in quantum-well wires. We use a dispersionless, macroscopic continuum model to describe the phonon confinement in quantum wires of square cross section. The polaron energy is calculated variationally incorporating the dynamic screening effects. We find that the confined phonon contribution to the polaron energy is comparable to that of bulk phonons in the density range N=104-107 cm-1. Screening effects within the random-phase approximation significantly reduce the electron-confined phonon interaction whereas the correlation effects tend to oppose this trend.

Optical-phonon transport and localization in periodic and Fibonacci polar-semiconductor superlattices

A transfer-matrix formalism is employed to study optical-phonon transport in a macroscopic continuum model for both periodic and Fibonacci polar-semiconductor superlattices. A phonon bandgap with subband structures is obtained for the periodic superlattices. However, in the Fibonacci superlattices, there is a spectrum trifurcation and self-similarity. The LO phonon localization length is calculated from which we confirm the existence of complete exponential localization of LO phonons.

Microscopic calculation of the electron-optical-phonon interaction in ultrathin GaAs/AlxGa1-xAs alloy quantum-well systems.

We have calculated electron--optical-phonon scattering rates in ultrathin GaAs/${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As alloy quantum-well systems of finite depth, based on a fully microscopic lattice dynamics approach for the phonon spectra. A pseudo-unit-cell model is utilized to calculate the lattice-dynamical properties of the ${\mathrm{Al}}_{\mathit{x}}$${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As alloy system together with the two-parameter Keating potential and the long-range Coulomb potential between ions. The polar interactions of quantum confined electrons with GaAs- and AlAs-like optical modes are presented for both intrasubband and intersubband transitions, and the comparison with other theoretical calculations made. Good agreement is obtained for the 1\ensuremath{\rightarrow}1 and 2\ensuremath{\rightarrow}1 transition rates between the microscopic pseuso-unit-cell model and a macroscopic two-pole dielectric continuum model.

Electron affinity of clusters and liquids of polar molecules: Theory of many-body polarization interactions

We examine the attractive interaction between an excess electron and the dipole moments of polar molecules forming a cluster or a liquid. This interaction determines the size dependence of the electron affinity of these clusters. We use the polarizable electropole model for the molecules and include interactions between the permanent and induced dipole moments. Thus, we obtain that the interaction between the excess electron and the molecules is strongly decreased by the induced polarization of the bound electrons, which is opposed to the permanent dipole moments and makes a dielectric screening. Note that previous calculations of the electron affinity of clusters of polar molecules resulted in much larger vertical detachment energies of the excess electron than observed in the experiment. This error is due to the use of an unphysical pair-potential approximation, which neglects the interaction between the permanent dipole moments and induced dipole moments. Thus the attractive interaction between the excess electron and the molecules and the resulting electronic binding energies have been overestimated. The results of our microscopic theory are identical with macroscopic continuum models and agree very well with experimental results. This theory also might be important for the study of solvent effects on oxidation–reduction reactions and on photosynthesis.

Polar optical vibrational modes in quantum dots.

A macroscopic continuum model coupling the mechanical vibrational amplitude and electrostatic potential is applied to obtain the optical vibrational modes in quantum-dot structures. A unified method of solution (valid for any type of nanostructure) that reduces the four coupled second-order differential equations to Helmholtz's and Laplace's equations is described. Analytical solutions for the vibrational amplitude and the Fr\"ohlich-type electron-phonon interaction are given for quantum dots with spherical geometry. The existence of surface modes and their relation to the matching boundary conditions are studied. A qualitative discussion of the ir and Raman activity of the calculated modes is given together with a comparison with the few existing experimental data.

Simplified microscopic model for electron-optical-phonon interactions in quantum wells.

A simplified microscopic model of optical phonons in dimensionally confined structures is formulated and applied to calculate electron-optical-phonon scattering rates in GaAs/AlAs quantum wells. For this simplified model which circumvents performing a complicated ab initio calculation of the force constants at the interface, it is demonstrated that the resulting dispersion relation and scattering rates for electron-optical-phonon interactions agree very well with those obtained from detailed ab initio studies. It is also shown that for GaAs/AlAs structures, the macroscopic dielectric continuum model provides a good approximation to the scattering rate predicted by the microscopic models

Micro-Raman scattering in ultrathin-layer superlattices: Evidence of zone-center anisotropy of optical phonons

Raman spectra of (001) (GaAs${)}_{\mathit{m}}$(AlAs${)}_{\mathit{n}}$, 1\ensuremath{\le}m, n\ensuremath{\le}7 superlattices have been measured in backscattering along x'\ensuremath{\parallel}[1\ifmmode\bar\else\textasciimacron\fi{}10], x\ensuremath{\parallel}[010], and z\ensuremath{\parallel}[001] directions with a microprobe. Confined longitudinal (LO) and transverse (TO) optical phonons with q either parallel or normal to the z axis have been studied in all the independent scattering geometries. A strong anisotropy is observed when the direction of the transferred momentum is changed from the z axis to x' or x. The most evident feature is that the Raman spectra in the z(x'x')z\ifmmode\bar\else\textasciimacron\fi{} geometry are dominated by ${\mathrm{LO}}_{1}$ confined phonons, whereas no signal at this energy is present in the y'(x'x')y\ifmmode\bar\else\textasciimacron\fi{} ' geometry. Modes evolving from ${\mathrm{TO}}_{1}$ and ${\mathrm{LO}}_{\mathit{n}}$ (n odd) phonons give rise to structures at intermediate frequencies in the x'(y'y')x\ifmmode\bar\else\textasciimacron\fi{} ' spectrum. In contrast to ${\mathrm{LO}}_{1}$, the measured ${\mathrm{TO}}_{1}$ energy remains constant, showing the isotropy of the lower-energy branch of this doubly degenerate mode. These results are explained by microscopic lattice-dynamical calculations performed in an ab initio scheme, which properly accounts for angular dispersion and mode mixing. We have also found that in the ultimate limit of confinement, when only one principal mode per branch can be considered, the phonon frequencies can be reproduced by the macroscopic dielectric continuum model, provided that the effect of phonon confinement is considered.

Efficient traffic flow simulation computations

Explicit numerical methods for solving macroscopic traffic flow continuum models have been studied and efficiently implemented in traffic simulation codes in the past. We studied and implemented implicit numerical methods for solving the flow conservation traffic model. We then wrote an experimental code in C simulating a freeway (un)congested pipeline and freeway entry/exit traffic flow. Tests with real data collected from the I-35 W freeway in Minneapolis were conducted on a workstation computer. The implicit methods gave the same (and in some cases better) accuracy as the Lax method. The implicit methods were (more than twice) faster than the Lax method.

Transition from LO-phonon to SO-phonon scattering in mesoscale structures

Macroscopic dielectric continuum models of optical-phonon modes predict enhancements in the magnitudes of the surface-optical (SO) modes in double-barrier heterostructures as the heterojunction-to-heterojunction separation is reduced. In this paper, the ratio of electron scattering by the SO-phonon modes to that by the (electrostatic) confined longitudinal-optical- (LO-) phonon modes is calculated for a GaAs/AlAs short-period superlattice based on the assumption that electron-SO-phonon scattering may be described by a scalar potential. The scaling of the ratio of electron-SO-phonon scattering to electron-LO-phonon scattering as a function of the superlattice period provides a sensitive test of the appropriateness of the scalar-potential model. The effect of phonon confinement on electron-optical-phonon scattering rates is presented for rectangular quantum wires as well. A major conclusion of these new results is that it is essential to model phonon confinement properly in predicting carrier transport properties in mesoscale structures.