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Controlling the thermal conductivity of semiconductors is of practical interest in optimizing the performance of thermoelectric and phononic devices. The insertion of inclusions of nanometer size in a semiconductor is an effective means of achieving such control; it has been proposed that the thermal conductivity of silicon could be reduced to 1 W/m/K using this approach and that a minimum in the heat conductivity would be reached for some optimal size of the inclusions. Yet the experimental verification of this design rule has been limited. In this work, we address this question by studying the thermal properties of silicon metalattices that consist of a periodic distribution of spherical inclusions with radii from 7 to 30 nm, embedded into silicon. Experimental measurements confirm that the thermal conductivity of silicon metalattices is as low as 1 W/m/K for silica inclusions and that this value can be further reduced to 0.16 W/m/K for silicon metalattices with empty pores. A detailed model of ballistic phonon transport suggests that this thermal conductivity is close to the lowest achievable by tuning the radius and spacing of the periodic inhomogeneities. This study is a significant step in elucidating the scaling laws that dictate ballistic heat transport at the nanoscale in silicon and other semiconductors.

Unlike conventional optics, plasmonics enables unrivaled concentration of optical energy well beyond the diffraction limit of light. However, a significant part of this energy is dissipated as heat. Plasmonic losses present a major hurdle in the development of plasmonic devices and circuits that can compete with other mature technologies. Until recently, they have largely kept the use of plasmonics to a few niche areas where loss is not a key factor, such as surface-enhanced Raman scattering and biochemical sensing. Here, we discuss the origin of plasmonic losses and various approaches to either minimize or mitigate them based on understanding of fundamental processes underlying surface plasmon modes excitation and decay. Along with the ongoing effort to find and synthesize better plasmonic materials, optical designs that modify the optical powerflow through plasmonic nanostructures can help in reducing both radiative damping and dissipative losses of surface plasmons. Another strategy relies on the development of hybrid photonic–plasmonic devices by coupling plasmonic nanostructures to resonant optical elements. Hybrid integration not only helps to reduce dissipative losses and radiative damping of surface plasmons, but also makes possible passive radiative cooling of nanodevices. Finally, we review emerging applications of thermoplasmonics that leverage Ohmic losses to achieve new enhanced functionalities. The most successful commercialized example of a loss-enabled novel application of plasmonics is heat-assisted magnetic recording. Other promising technological directions include thermal emission manipulation, cancer therapy, nanofabrication, nanomanipulation, plasmon-enabled material spectroscopy and thermo-catalysis, solar water treatment, and thermophotovoltaics., United States. Department of Energy. Office of Basic Energy Sciences (Grant DE-FG02-02ER45977), United States. Defense Advanced Research Projects Agency (Grant DE-AR0000471), Solid-State Solar-Thermal Energy Conversion Center (Award DE-SC0001299), Solid-State Solar-Thermal Energy Conversion Center (Award DE-FG02-09ER46577)

Despite the established knowledge that crystal dislocations can affect a material’s superconducting properties, the exact mechanism of the electron-dislocation interaction in a dislocated superconductor has long been missing. Being a type of defect, dislocations are expected to decrease a material’s superconducting transition temperature (Tc) by breaking the coherence. Yet experimentally, even in isotropic type I superconductors, dislocations can either decrease, increase, or have little influence on Tc. These experimental findings have yet to be understood. Although the anisotropic pairing in dirty superconductors has explained impurity-induced Tc reduction, no quantitative agreement has been reached in the case a dislocation given its complexity. In this study, by generalizing the one-dimensional quantized dislocation field to three dimensions, we reveal that there are indeed two distinct types of electron-dislocation interactions. Besides the usual electron-dislocation potential scattering, there is an...

We show the existence of a space-charge effect in the measurement of the low-temperature Seebeck coefficient in insulators. The Seebeck coefficient $$ S(T)$$ is obtained by measurement of the voltage $$\Delta V$$ due to a temperature gradient $$(\Delta T, S=-\Delta V/\Delta T)$$ . The space-charge effect makes the voltage go to zero in insulators, even if the Seebeck coefficient does not vanish. We propose that the Seebeck coefficient does not actually vanish in insulators, contrary to common belief. We also propose that variable-range hopping is not observed in conductivity measurements.

We discuss the theory behind low-temperature measurement of the Seebeck coefficient in materials with few electrons, such as insulators and lightly doped semiconductors. The Seebeck coefficient is defined thermodynamically as being related to the electrochemical potential. However, the measurement is of the voltage for a material with no current, which is not the same as the gradient of the electrochemical potential.

We propose a quantum model to explain the dynamic polarization observed above the transition temperature of BaTiO3 and below the Burns temperature. Our theory also applies to other perovskite ferroelectrics.

We calculate the response of a free electron metal to a static charge Q located outside of the metal surface. We derive the electrostatic potential outside of the surface, inside of the surface, and the frequency response. Our derivation uses standard quantum mechanics, and the results are found exactly in the random phase approximation. Unlike prior calculations, which made approximations, our results do not agree with classical image theory.

A theory is presented for the thermal boundary resistance, usually called the Kapitza resistance, between a planar interface that separates a solid and a fluid. The theory includes transverse modes in the fluid that can carry diffusive heat from the surface into the fluid volume.

Raman scattering from surface optic (SO) phonons has been observed and identified in cylindrical and rectangular cross-section nanowires with lateral dimensions in the range 20-50 nm and lengths of ~ 10 microns. The position of the SO band is found to depend on both the shape of the wire cross section (i. e., circular or rectangular) and on the dielectric constant of the external medium. The position of the SO band in GaP and ZnS nanowires is in good agreement with a dielectric continuum model that takes the shape of the wire cross section into account. We propose that the symmetry breaking mechanism which activates the Raman sacttering of the SO phonon in our nanowires is a quasi-periodic modulation of the cross-sectional area along the nanowire axis. We suggest this modulation stems from an interesting growth instability associated with vapor-liquid-solid (VLS) growth.

State-of-the-art infrared (IR) thermal imaging was used to monitor the heating of ZnO varistors by electrical transients. On a macroscopic scale (e.g., 10 mm), heating in large varistor blocks (i.e., diameter of 42 mm) was found to be the greatest near the block edges and to be approximately radially symmetric in blocks fabricated at a low aspect ratio. In blocks fabricated at a higher aspect ratio, the heating was less symmetric, presumably because uniform properties are more difficult to achieve. Nonuniform heating in large blocks can be attributed to processing-induced variations in the electrical properties of the blocks. On an intermediate size scale (e.g., 1 mm), the heating in small varistor disks (e.g., diameter of 10 mm) was observed to be most intense along localized electrical paths. The high electrical conductivity of these paths originates from the statistical fluctuations in properties that inevitably occur in polycrystalline materials. On a microscopic scale (e.g., 10 μm), the heating in thin varistor slices (e.g., thickness of 100 μm) was observed to be localized in strings of tiny hot spots. The hot spots occur at the grain boundaries in a conducting path, where the potential is decreased across Schottky-type barriers and the heat is generated. The experimentally observed heating is interpreted by applying transport theory and using computer simulations. It is shown that, on the scale of the grain size, the heat transfer is too fast to permit temperature differences that could cause a varistor failure. Current localization and nonuniform heating on an intermediate size scale can have a microstructural origin (e.g., statistical fluctuations of grain sizes and grain-boundary properties). However, these are shown to be significant only in small varistors, whereas destructive failures (puncture and cracking) of large varistor blocks can be caused only by nonuniform heating on a macroscopic scale.

Raman scattering studies of crystalline GaP nanowires reveal a strong additional peak in the first order spectrum that can be clearly assigned to surface optical (SO) phonons. The frequency of this SO peak is found to be sensitive to the dielectric constant of the surrounding medium in which the nanowire is embedded. A theory for the SO phonons in cylindrical wires is presented, the SO mode dispersion ω(q) and the experimental peak frequency are then used to predict the wavelength of the dominant Fourier component of the surface potential responsible for activating the SO mode. Interestingly, this SO phonon wavelength is found to agree with the wavelength of diameter modulation observed for some nanowires.

Rapid progress in the synthesis and processing of materials with structure on nanometer length scales has created a demand for greater scientific understanding of thermal transport in nanoscale devices, individual nanostructures, and nanostructured materials. This review emphasizes developments in experiment, theory, and computation that have occurred in the past ten years and summarizes the present status of the field. Interfaces between materials become increasingly important on small length scales. The thermal conductance of many solid–solid interfaces have been studied experimentally but the range of observed interface properties is much smaller than predicted by simple theory. Classical molecular dynamics simulations are emerging as a powerful tool for calculations of thermal conductance and phonon scattering, and may provide for a lively interplay of experiment and theory in the near term. Fundamental issues remain concerning the correct definitions of temperature in nonequilibrium nanoscale systems. Modern Si microelectronics are now firmly in the nanoscale regime—experiments have demonstrated that the close proximity of interfaces and the extremely small volume of heat dissipation strongly modifies thermal transport, thereby aggravating problems of thermal management. Microelectronic devices are too large to yield to atomic-level simulation in the foreseeable future and, therefore, calculations of thermal transport must rely on solutions of the Boltzmann transport equation; microscopic phonon scattering rates needed for predictive models are, even for Si, poorly known. Low-dimensional nanostructures, such as carbon nanotubes, are predicted to have novel transport properties; the first quantitative experiments of the thermal conductivity of nanotubes have recently been achieved using microfabricated measurement systems. Nanoscale porosity decreases the permittivity of amorphous dielectrics but porosity also strongly decreases the thermal conductivity. The promise of improved thermoelectric materials and problems of thermal management of optoelectronic devices have stimulated extensive studies of semiconductor superlattices; agreement between experiment and theory is generally poor. Advances in measurement methods, e.g., the 3ω method, time-domain thermoreflectance, sources of coherent phonons, microfabricated test structures, and the scanning thermal microscope, are enabling new capabilities for nanoscale thermal metrology.

Electrical conductivity, thermopower, and lattice strain were investigated in the tetragonal tungsten bronze structured $({\mathrm{Sr}}_{x},{\mathrm{Ba}}_{1\ensuremath{-}x}){\mathrm{Nb}}_{2}{\mathrm{O}}_{6\ensuremath{-}\ensuremath{\delta}}$ system for $0.7gxg0.4$ with large values of \ensuremath{\delta}. These materials show attractive thermoelectric characteristics, especially in single-crystal form. Here, the Sr/Ba ratio was changed in order to vary the material between a normal ferroelectric with long-range polarization to relaxor behavior with short-range order and dynamic polarization. The influence of this on the electrical conduction mechanisms was then investigated. The temperature dependence of both the thermopower and differential activation energy for conduction suggests that the electronic conduction is controlled by an impurity band with a mobility edge separating localized and delocalized states. Conduction is controlled via hopping at low temperatures, and as temperature rises electrons are activated above the mobility edge, resulting in a large increase in electrical conductivity. For relaxor ferroelectric-based compositions, when dynamic short-range order polarization is present in the system, trends in the differential activation energy and thermopower show deviations from this conduction mechanism. The results are consistent with the polarization acting as a source of disorder that affects the location of the mobility edge and, therefore, the activation energy for conduction.

A fast current pulse causes a material to heat and undergo rapid expansion. We calculate the response of a cylindrically shaped material to a pulse on the microsecond time scale. The first step is to obtain the breathing modes of elastic oscillation of the cylinder. These modes are calculated using a Rayleigh–Ritz variational method introduced by Demarest for cubes. The boundary conditions are derived, which give the amplitude of each elastic mode in response to the sudden heating. The results are illustrated by calculations on a station arrester made of a ZnO ceramic.

A simple thermo-mechanical model is applied to evaluate the influence of the nonuniformity of ZnO varistor disks used in surge arresters on their energy handling capability. Puncture is the dominating failure mode for slightly nonuniform disks, but cracking becomes more likely as the degree of nonuniformities increases. It is shown that minimization of the chance of a failure of varistor disks at high-current pulses can be achieved by adjusting their resistivity in the upturn region of the I-V characteristic. Simulation of the behavior of varistor disks under high-current 4/10 /spl mu/s pulses required by the ANSI standard tests shows that these tests provide very little information about the actual energy handling capability of the disks. This conclusion suggests that alternate test methods should be developed and included in the relevant standards.

Opals are structures composed of the closed packing of spheres in the size range of nano-to-micro meter. They are sintered to create small necks at the points of contact. We have solved the diffusion problem in such structures. The relation between the diffusion coefficient and the termal and electrical conductivity makes possible to estimate the transport coefficients of opal structures. We estimate this changes as function of the neck size and the mean-free path of the carriers. The theory presented is also applicable to the diffusion problem in other periodic structures., Comment: Submitted to PRB

Equations are solved to give the variations in temperature, density, and potential, when a solid has electrical currents, heat currents, and particle diffusion. Solutions are presented in one dimension for currents down a bar. These solutions are used to calculate the efficiency of a thermoelectric refrigerator, which is optimized to give the coefficient of performance (COP). The COP depends upon the temperature difference ΔT, but does not depend upon the density difference Δn between the two ends of the bar.

The electron-phonon interaction in low-dimensional tight-binding systems is discussed. A sheet of graphite, which is two-dimensional, and an armchair single-wall carbon nanotube (SWNT), which is quasi-one-dimensional, are taken as examples. For the modulated hopping the matrix elements for both systems are derived in the context of a two parameter model for the phonon vibrational spectrum. It is found that they (for both structures) display a deformation type of potential, and are reduced by a factor of $(1\ensuremath{-}R),$ where R depends on the phonon parameters. It is also shown that the ordinary electron-phonon coupling displays a deformation type of approximation for both systems. Next, a different type of interaction is considered---the phonon modulated electron-electron interaction. It gives two contributions---random phase approximation with one phonon line and exchange interaction with one phonon line. We find that for the two-dimensional (2D) graphene and for the quasi-1D (10,10) SWNT, the modulated hopping and exchange coupling govern the electron transport at room temperatures.

We analyze the influence of defects on conductance, density of states, and localization in (N{sub a},N{sub a}) armchair carbon nanotubes within a tight-binding model. Using the transfer-matrix method, we calculate the reflection (related to the conductance) from a sequence of defects and relate its energy dependence near the Fermi level to the appearance of a quasibound state. This state is also seen in the density of states and in the energy dependence of the quasiparticle lifetime. We compute the localization length {xi}({omega}) as a function of energy {omega}. Comparison of {xi}(0) with the mean free path l{sub mfp} in the limit of small defect concentration c and small defect strength E leads to a simple approximate relation {xi}(0){approx}3l{sub mfp}=3{times}3aN{sub a}t{sup 2}/2cE{sup 2} (t{emdash} hopping integral, a{emdash} lattice constant). {copyright} {ital 1999} {ital The American Physical Society}

The phonon modulation of electron-electron interactions are calculated for solids described by tight-binding models. In some cases the interaction can be larger than the usual electron-phonon effects.

Tight-binding calculations are presented of the ground state energies of metal atoms absorbed on the surface of the same or similar metals. We consider rows and clusters of adsorbate, and find significant energy variations as a function of the number of the atoms. Results are presented for the (001) face both for fcc and bcc crystals. General results that are applicable for any elemental fcc or bcc crystal are obtained in the present paper. We include only s-wave tight-binding orbits in the nearest neighbor approximation.

We use a transfer-matrix method to study defects in a tight-binding model of carbon nanotubes. We calculate the reflection coefficient {ital R} for a simple barrier created by a pointlike defect of strength {ital E} in armchair (N{sub a},N{sub a}) and zigzag (N{sub a},0) nanotubes for the whole range of energy {omega} and arbitrary number of conducting channels. We find that {ital R} scales at the Fermi level (i.e., {omega}=0) as R=s(E/t){sup 2}/N{sub a}{sup 2} (t being the hopping parameter), where s{approx}1/6 (for the armchair nanotubes) and s{approx}1/2 (for the zigzag nanotubes). We also perform a similar calculation for a {open_quotes}5-77-5{close_quotes} defect and find the results to be like the ones obtained for a strong point defect with E=6t. {copyright} {ital 1998} {ital The American Physical Society}

Thermal and mechanical behavior of ZnO varistors used in surge arresters is simulated for current pulses of various magnitude and duration. Time dependence of the temperature profile and of the distribution of thermal stresses is computed by solving heat transfer equations for varistor disks with nonuniform electrical properties. The simulations identify failure processes and determine energy absorption capability of varistor elements. The results are in excellent agreement with experiment and provide fundamental explanation of the energy handling dependence upon current surge intensity and duration.

We report calculations which show that the band structure of CoSb{sub 3} is typical of a narrow-band-gap semiconductor. The gap is strongly dependent on the relative position of the Sb atoms inside the unit cell. We obtain a band gap of 0.22 eV after minimization of these positions. This value is more than four times larger than the result of a previous calculation, which reported that the energy bands near the Fermi surface are unusual. The electronic states close to the Fermi level are properly described by a two-band Kane model. The calculated effective masses and band gap are in excellent agreement with Shubnikov{endash}de Haas and Hall effect measurements. Recent measurements of the transport coefficients of this compound can be understood assuming it is a narrow-band-gap semiconductor, in agreement with our results. {copyright} {ital 1998} {ital The American Physical Society}

A new method of refrigeration is proposed. Efficient cooling is obtained by thermionic emission of electrons over Schottky barriers between metals and semiconductors. Since the barriers have to be thin, each barrier can have only a small temperature difference $(\ensuremath{\sim}1\mathrm{K})$. Macroscopic cooling is obtained with a multilayer device. The same device is also an efficient generator of electrical power. A complete analytic theory is provided.

A new method of refrigeration is proposed. Cooling is obtained by thermionic emission of electrons over periodic barriers in a multilayer geometry. These could be either Schottky barriers between metals and semiconductors or else barriers in a semiconductor superlattice. The same device is an efficient power generator. A complete theory is provided., Comment: 17 pages with 5 postscript figures, submitted to J. Appl. Phys

We predict in Si/Ge superlattices a dramatic high-temperature suppression of the perpendicular thermal transport. Total internal reflection confines the superlattice modes and significantly reduces the average group velocity at nonzero in-plane momenta. These consequences of the acoustic mismatch cause at high temperatures an order-of-magnitude reduction in the ratio of superlattice thermal conductivity to phonon relaxation time.

The single-site Anderson model with degeneracy ${N}_{f}$ is solved for the case in which the lowest eigenstate contains $L$ electrons in the $f$ orbital, where $0lLl{N}_{f}.$ The noncrossing approximation is used to find the density of states including the Kondo resonance. The Seebeck coefficient is calculated as a function of temperature for every value of $L$. We find that the Seebeck coefficient is largest for $L=1$ and declines in value as $L$ increases. This suggests that large Seebeck coefficients are not possible in mixed-valent systems besides those of cerium or ytterbium.

The Eliashberg equations are solved for the transition temperature of a superconductor for the case that the phonons are harmonic but the electron-phonon interaction is anharmonic. The anharmonic interaction increases the transition temperature for the same coupling strength. Reentrant behavior is predicted for large anharmonicity. {copyright} {ital 1997} {ital The American Physical Society}

We report molecular dynamics simulations of nonequilibrium heat flow in a solid system in the local-equilibrium or hydrodynamic approximation and demonstrate that local equilibrium can be achieved in small numbers of atomic layers over long simulation runs. From the dynamical simulations, we calculate the thermal boundary (Kapitza) resistance that arises from heat flow across Si grain boundaries and compare with the traditional approach based on calculating the transmission and reflection of harmonic phonons at the grain boundary.

Thermoelectrics is an old field. In 1823, Thomas Seebeck discovered that a voltage drop appears across a sample that has a temperature gradient. This phenomenon provided the basis for thermocouples used for measuring temperature and for thermoelectric power generators. In 1838, Heinrich Lenz placed a drop of water on the junction of metal wires made of bismuth and antimony. Passing an electric current through the junction in one direction caused the water to freeze, and reversing the current caused the ice to quickly melt; thus thermoelectric refrigeration was demonstrated (figure 1).

Electrical transport in zinc oxide varistors is simulated using two‐dimensional Voronoi networks. The networks are assumed to contain randomly distributed grain boundaries of three electrical types: (1) high nonlinearity (i.e., ‘‘good’’) junctions; (2) poor nonlinearity (i.e., ‘‘bad’’) junctions; and (3) linear with low‐resistivity (i.e., ohmic) junctions. These type classifications are those found in experimental measurements. By varying the type concentrations, the simulated current density versus electric field (J–E) characteristics can be made to conform to the different experimentally observed characteristics of ZnO varistors. These characteristics include the sharpness of switching at the transition between ohmic and nonlinear J–E response (i.e., knee region), as well as the degree of nonlinearity. It is shown that the reduction of the nonlinearity coefficient of bulk varistors, relative to that of isolated grain boundaries, can be explained only by the presence of ‘‘bad’’ varistor junctions.

The anharmonic electron-phonon problem is solved in the infinite-dimensional limit using quantum Monte Carlo simulation. Charge-density-wave order is seen to remain at half filling even though the anharmonicity removes the particle-hole symmetry (and hence the nesting instability) of the model. Superconductivity is strongly favored away from half filling (relative to the charge-density-wave order) but the anharmonicity does not enhance transition temperatures over the maximal values found in the harmonic limit., 5 pages typeset in ReVTeX. Four encapsulated postscript files included

Carbon-filled polyethylene composites were fabricated and tested to establish the practical lower limit of their electrical resistivity at room temperature and to investigate the trade-offs between low resistivity and the magnitude of the resistance anomaly (i.e., a large positive temperature coefficient of resistivity) that appears when such composites are heated through the polyethylene crystalline melting transition. Carbon blacks with large particle size and low surface area provided low-resistivity composites having large resistance anomalies. The largest resistance anomalies were found in composites that were well mixed, but the room-temperature resistivity also increased in composites that were cycled repetitively throught he crystalline-melting transition. A mixture of carbon blacks of two different sizes provided a lower resistance than was found in a material with the same fill of only the coarser black. By controlling the composition and the processing, composites were made with room-temperature resistivities lower than 0.2 ohm cm and resistance changes of at least 2 orders of magnitude. A resistance change of as much as 5 orders of magnitude was obtained for composites with room-temperature resistivities of only 1 ohm cm. {copyright} {ital 1996 Materials Research Society.}

The quantum Boltzmann equation is derived for photons. The form of the scattering and absorption terms for the case of photon diffusion is discussed in detail. We show how the structure factor of the scattering centers enters into the scattering rate of the photons.

What electronic structure provides the largest figure of merit for thermoelectric materials? To answer that question, we write the electrical conductivity, thermopower, and thermal conductivity as integrals of a single function, the transport distribution. Then we derive the mathematical function for the transport distribution, which gives the largest figure of merit. A delta-shaped transport distribution is found to maximize the thermoelectric properties. This result indicates that a narrow distribution of the energy of the electrons participating in the transport process is needed for maximum thermoelectric efficiency. Some possible realizations of this idea are discussed.

Thermal and mechanical behavior of high power ZnO surge arresters under current pulses of various magnitude and duration is simulated. By solving heat transfer equations for a varistor disk with nonuniform electrical properties, we compute the time dependence of the temperature profile and the distribution of thermal stresses. The simulations identify failure processes and determine energy handling capability of varistor elements as a function of the applied current. The results are in good agreement with the experimental data and explain the energy handling dependence upon the surge intensity and duration.

A realistic model of transport properties of zinc oxide varistors is constructed from two‐dimensional Voronoi networks and studied via computer simulations. In agreement with experimental microcontact measurements made on individual junctions, the networks are assumed to contain randomly distributed microjunctions of two types: (1) electrically active with highly nonlinear current‐voltage (I‐V) characteristics and (2) ohmic, i.e., with linear I‐V characteristics. Effects of the ohmic grain boundaries in the network are simulated for various concentrations and resistivities. Shapes of the simulated I‐V characteristics and current dependence of the coefficient of nonlinearity of the network are in good agreement with those experimentally observed for thin varistor samples and in the measurements employing various surface electrode patterns. It is found that the breakdown voltage of the networks increases with the number of the ohmic grain boundaries, except when their resistivity is so low that it becomes c...

We present calculations for the supercooling of a thermoelectric material during a transient current pulse. At room temperature, a standard steady-state cooler produces a maximum temperature drop of 82 K. During a current pulse, this value can be increase to about 108 K. While prior calculations focused on the optimization of the current pulse with respect to length and height, we investigate the influence of the pulse shape upon the cooling mechanism. Our results show that using a quadratic pulse form, the supercooling effect can be improved and a maximum temperature drop of 116 K can be achieved, exceeding all previously reported results by 8 K. The mechanism increases the ratio of ZTeff∕ZT from the previous value of 1.76 to a maximum of 2.01. The optimized pulse shape requires less energy and, therefore, prevents extensive heating of the material after the minimum temperature is reached, resulting in an increase of efficiency by approximately a factor of 2.

Boltzmann’s equation is solved for the diffusion of nonabsorbing particles through a thick slab. The dimensionless thickness D=dμs, where d is the thickness and μs is the rate of scattering. Our solution neglects terms of order O(e−D) and is exact in the limit that D≫1. Simple expressions are obtained for the reflection and transmission coefficients, as well as for the angular distributions on the front and back sides of the slab.

The Hartree-Fock (HF) exchange energy of the two-dimensional electron gas is calculated at nonzero temperatures. The calculation is done self-consistently in that the HF self-energy is included self-consistently in the Fermi-Dirac functions. We also calculate the first and second derivatives of the exchange energy with respect to density, which are useful for density-functional calculations at nonzero temperatures. These calculations may be useful in understanding the temperature dependence of two-dimensional systems such as electrons trapped two dimensionally in semiconductor interfaces.

The anisotropic electron-phonon interaction is used to calculate the transition temperature ${\mathit{T}}_{\mathit{c}}$ of a superconductor. We derive constraints on the maximum value of the anisotropic interaction, and use these extremal values to calculate the maximum increase in ${\mathit{T}}_{\mathit{c}}$ from the anisotropy. Calculations are done for electrons on a two-dimensional square lattice using an Einstein model for phonons.

An impurity in a conductor will cause oscillations in the charge density near the impurity. We calculate these Friedel Oscillations for impurities on a square lattice while using the tight-binding model. Far from the impurity the oscillations are described by a set of phase shifts. These are the same phase shifts which enter the Friedel Sum Rule.