Conventional Transport Theory Research Articles

Large-bipolaron transport and cuprate superconductors.

The in-plane electronic transport of large bipolarons that move within parallel conducting sheets is considered. The Coulomb interactions between the heavy bipolarons in this planar system produce a distinctive spectrum for collective excitations. Transport of the quasiparticles associated with the collective excitations is treated via the conventional Boltzmann transport theory. However, the scattering of the heavy quasiparticles by acoustic phonons is taken to be analogous to the distinctive scattering of individual large polarons. In particular, as with strong-coupling (adiabatic) large polarons, the scattering time is taken to be proportional to the quasiparticle effective mass. The resistivity, the Hall effect, and the thermoelectric power are determined with this scheme. With a constant density of carriers, the resistivity is proportional to the temperature. The Hall number, the number of carriers deduced from the reciprocal of the Hall constant, exceeds the true carrier density by a ratio that rises linearly with temperature. In addition, the thermoelectric power falls in magnitude and ultimately changes sign as the temperature is raised. These unusual transport properties, as well as the ir conductivity, are consistent with the observed behaviors of the high-${\mathit{T}}_{\mathit{c}}$ cuprate superconductors.

Diverging resistivity anisotropy with decreasing temperature in 60-K YBa2Cu2O7-y.

The in-plane resistivity ${\ensuremath{\rho}}_{\mathrm{ab}}$ of 60-K ${\mathrm{YBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{y}}$ is equal to (340\ifmmode\pm\else\textpm\fi{}30) \ensuremath{\mu}\ensuremath{\Omega} cm at 290 K and varies approximately linearly with temperature T. The anisotropy ${\ensuremath{\rho}}_{c}$/${\ensuremath{\rho}}_{\mathrm{ab}}$ increases from \ensuremath{\sim}100 to 2000 as T decreases from 290 to 80 K. In the four crystals studied ${\ensuremath{\rho}}_{c}$ is observed to increase monotonically with decreasing T. This is an intrinsic property of the normal state which has not been explained by conventional transport theory.

Theory of conductivity in superlattice minibands.

We have calculated the impurity-scattering--limited electrical conductivity for vertical transport in superlattice minibands. For sufficiently small carrier density and/or ``large'' disorder the collisional broadening \ensuremath{\Gamma}(\ensuremath{\mu}) can be larger than the chemical potential \ensuremath{\mu}. In such situations the quasiparticle approximation breaks down, and use of the conventional Bloch-Boltzmann transport theory is unreliable. Also, as the period of the superlattice increases, the ratio \ensuremath{\Gamma}(\ensuremath{\mu})/\ensuremath{\mu} grows, resulting in a reduction of the mobility, leading eventually to Anderson localization. In addition, the carriers in the miniband become nondegenerate already at low temperatures, giving rise to a significant temperature dependence of the mobility. Furthermore, due to the unique shape of the Fermi surface of the superlattice the mobility becomes independent of the carrier density when the chemical potential exceeds the miniband width.

A new procedure for determining the diffusion coefficients of electron swarms according to the modern transport theory

In this paper we are concerned with the solution of the kinetic equations which are met when expanding the electron velocity distribution function with respect to the gradient of the electron density in the Boltzmann equation. The approximation of the corresponding zero- and first-order coefficients by an expansion in Legendre, or associated Legendre, polynomials up to a given number (2l)_ of terms, which transform the kinetic equations into three coupled hierarchies for the corresponding expansion coefficients, is assumed. The solutions of these hierarchies permit to determine the drift speed, as well as the longitudinal and transverse diffusion coefficients of an electron swarm. Based on an analysis of the mathematical structure of these singular ordinary differential equation systems (with additional difference terms due to the inelastic,i.e. energy loss, collision processes) a new procedure is developed which isolates and constructs (for arbitrary even approximation order 2l) the nonsingular part of the general solutions (NSPGS) of each hierarchy. The NSPGS is firstly obtained in the region of small electron energies (up to an appropriate connection point) and secondly in the region of large energies (down to the connection point) starting from the two singular points of each hierarchy,i.e. from very small and sufficiently large energies, respectively. The continuous connection of the two NSPGSs, obtained in the two different energy regions, together with the normalization condition, yield the physically relevant solution for each one of the three hierarchies and make it possible to calculate the diffusion coefficients we are particularly interested in. Our new technique, even if much more complex, is a logical generalization of the conventional backward prolongation technique used for the well-known two-term approximation of the stationary and spatially homogeneous Boltzmann equation. The present paper also reports a first application of this new procedure to a model plasma. The diffusion coefficients are calculated with increasing approximation order 2l up to their converged values, under different parameter conditions of the model and are compared with corresponding values found by accurate Monte Carlo simulations as well as with results of the conventional transport theory for the same model. The diffusion coefficients obtained for different (even) order of approximation well illustrate the power of our new procedure. In particular, the converged values of the diffusion coefficients are found to be in very good agreement with the corresponding Monte Carlo values.

Longitudinal- and transversal-diffusion coefficients for electrons in CH4, Ar and CO2

This paper reports the results of calculations of diffusion coefficients of electron swarms in an electric field, performed according to the modern transport theory. The results are compared with corresponding values obtained by Monte Carlo simulations as well as by calculations performed according to the conventional transport theory for the three gases CH4. Ar and CO2. For what concerns the calculation of the longitudinal- and transverse-diffusion coefficients, use is made of a new procedure we have recently developed. The procedure solves by Legendre and associated Legendre polynomial expansions the kinetic equations which are obtained for the hydrodynamic state gy expanding the electron velocity distribution function with respect to the gradient of the electron density in the Boltzmann equation. The solutions of the kinetic equations are obtained with increasing (even) approximation order of the Legendre and associated Legendre polynomial expansion up to convergence.

The response of a tunnelling particle to an impulse

A classical particle moving in a one-dimensional periodic potential of height V vibrates if its energy E is less than V and translates if E>V. It responds to an impulse by a change in its mean velocity only if E>V. Due to tunnelling, the quantum states of a particle do not divide sharply into vibrational and translational ones. The author considers the question of whether a quantum particle with E<V responds to an impulse (due to lattice fluctuations) by subsequently increasing its rate of tunnelling in the direction of the impulse and diminishing it in the opposite direction. (The author calls this response driven tunnelling). The question is relevant to all kinds of motion in crystal lattices. It is shown that where conventional transport theory ignores driven tunnelling, it is inconsistent with classical mechanics. A simple theory of driven tunnelling is described in the context of methyl group rotation. It leads to a Hamiltonian which has both a scalar and a vector potential. The latter is changed by an impulse and is responsible for the change in the subsequent tunnelling behaviour.

Stopping of slow recoil atoms in gases

Nuclear Resonance Fluorescence spectra contain information on the slowing-down behaviour of recoiling atoms in the energy range pertinent to the specific reaction. On the basis of conventional transport theory, measured spectra for arsenic atoms with energies ≲15 eV slowing down in helium are analysed. The assumption of continuous slowing down turns out to be well justified, and the thermal motion of the gas atoms is shown to influence the effective stopping power only slightly. Somewhat surprising in view of uncertainties concerning the charge state of the recoiling atoms, theoretical predictions based on Lenz-Jensen elastic interaction agree well with the measured spectra. At present, the experimental data do not allow direct inversion, but the calculated spectra are shown to depend sensitively on the stopping-power input.

Stochastic equations of particle transport scattering theory

The Pal-Bell stochastic regeneration equation is adapted to the special case of neutron scattering with capture (no fission) in the transport model, specifying the density of the probabilities of finding exactlyn neutrons distributed throughout the system. It is shown that this form may be regarded as the Kolmogorov backward equations for the Markov problem and the equivalent forward equations are obtained from adjoint-operator theory. The general solution is developed in terms of a conventional Boltzmann transport theory solution and the nature of the moment equations is deduced. Finally the problem is cast as a variational principle.

The diffusion and drift of electrons in gases: II. A Monte-Carlo simulation in argon

A Monte-Carlo analysis of the motion of electrons in argon is presented to test the conventional electron transport theory and, in particular, the debated theories of parallel diffusion in a gas which lends itself very well to this end. The values of the ratio E/ N between field intensity and atom number density are always so low that a high percentage of the electrons is in the energy region where discrepancies between Monte Carlo and theory may reasonably be expected. It is found, on the contrary, that the agreement is always very good for any E/ N (i.e. within 3–4%) even for a cross section so strongly dependent on the electron energy as that recommended by Golden. The theories of parallel diffusion by Parker and Lowke, Huxley and, particularly, by Skullerud are shown to give accurate values of the diffusion coefficient D ‖ even in cases where more than 80% of the electrons is in the (low) energy region, where the cross section is strongly decreasing with ϵ. This conclusion is hoped to clarify definitively the discrepancies which have been so disturbing many people in the last years.

Thermopower behaviour of amorphous versus crystalline Ge and GeTe films

In continuation of our studies of the structural, optical and electrical properties of amorphous (a-) versus crystalline (c-) Ge and GeTe films, the present paper reports the temperature dependence of the thermoelectric power (α) of these films relative to evaporated gold films. In c-films, α increases nearly linearly with temperature below about 300°K (metallic or degenerate semiconducting behaviour) and then rises rapidly to a saturation value. The linear temperature dependence is characteristic of metals or degenerate semiconductors. The rapid rise is associated with rapid changes of mobility in the degenerate c-films. The thermoelectric data of GeTe films support our earlier conclusion that only a single valence band contributes to the transport process, and further, that the maximum of the valence band occurs at k ≠ 0. The thermoelectric data of a-films bear no correlation with those of c-films. For a-GeTe films, a constant and positive value of α∼0.88 mV/°K is observed in the measurable temperature range 175°–375°K. In a-Ge films, α is negative below 300°K with a constant value of about 0.08 mV/°K below ∼250°K. It reverses sign above 300°K and rises rapidly to a value of about 0.96 mV/°K at ∼420°K and then decreases above 420°K, most probably due to alloying at the Au-Ge interface. The behaviour of thermopower in amorphous films cannot be understood by conventional transport theories. A qualitative interpretation is possible on the basis of heat transport by a hopping process and a speculative energy band diagram.

Conduction band symmetry and spin-dependent polaron transport in the alkali halides

The transverse magnetoconductivity of photo-excited electrons in KCl, KBr, KI, NaI and RbCl has been studied at low temperatures as a function of the orientation of the magnetic field with respect to the crystal axes. The minimum of the lowest conduction band in these substances is found to be highly isotropic. The first studies are reported of the dependence on the orientation of the conduction electron spins of electronic transport in alkali halide crystals containing F centres. These measurements can be understood in terms of the interaction of spin polarized conduction electron with spin polarized F centres. It appears that the transport of photoelectrons at 4.2 degrees K in KCl, KBr and KI containing 1016 to 1017 F centres per cm3 occurs by a diffusive motion as in conventional transport theory and that a spin-dependent scattering or trapping time may be defined.

Electric Properties ofn-Type Liquid Alloys of Thallium and Tellurium

Measurements have been made of the electrical conductivity $\ensuremath{\sigma}$ and Seebeck coefficient $S$ of Tl-Te liquid alloys as a function of the composition $X$ (at.% Tl) and temperature $T$ for $67&lt;X&lt;72$. Also, the effects of doping the $n$-type binary solution with relatively small amounts of a third element were studied, with Ag, Cd, In, Sn, or Sb as the third element. In most ranges of $T$ and $X$, there was little dependence of $\ensuremath{\sigma}$ and $S$ on $T$. It was possible to fit isothermal data to conventional transport theory in a rather comprehensive way. Plots of all data for $\ensuremath{\sigma}$ versus $S$, including solutions containing a third element, fell on a single theoretical curve based on Fermi-Dirac integrals, corresponding to $\mathcal{r}=\ensuremath{-}\frac{1}{2}$, viz., $\ensuremath{\tau}={\ensuremath{\tau}}_{\ensuremath{\perp}}{\ensuremath{\epsilon}}^{r}$, where $\ensuremath{\tau}$ is the scattering time and $\ensuremath{\epsilon}$ is the kinetic energy of the electrons. This value of $\mathcal{r}$, together with the finding that ${\ensuremath{\tau}}_{1}$ is independent of $T$, indicates that scattering is caused by the intrinsic (nonthermal) disorder of the liquid. Comparison of the experimental dependence of $S$ on the composition of Tl-Te solutions, with the theoretical dependence of the electron concentration $n$ on $S$, shows that $n$ is proportional to the concentration of Tl in excess of the composition ${\mathrm{Tl}}_{2}$Te. Use of the same theoretical curve for the dependence of $S$ on the concentration $A$ of the various doping elements showed again a linear dependence. Comparison of the relative slopes $\frac{\mathrm{dn}}{\mathrm{dA}}$ for the different elements indicates that one could assign integral valences as follows: Tl, +3; Ag, +1; Cd, +2; In, +1; Sn, +2; Sb, -1. Consideration of other aspects of transport theory, particularly the relative magnitudes of the de Broglie wavelength and the scattering distance, suggests that a large fraction of the electron charge in the conduction band is localized as the result of screening of the potential of the donor and acceptor ions. Solutions of the nonlinear Thomas-Fermi screening equation indicate that physically reasonable magnitudes of the effective dielectric constant $K$ leads to value of $\ensuremath{\alpha}$, the fraction of free-electron charge, which are consistent with the range of values required by the transport parameters. We deduce that $0.15\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{&lt;}\ensuremath{\alpha}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{&lt;}0.5$, and the effective-mass ratio $\frac{{m}^{*}}{m}$ lies between 1.0 and 2.0, corresponding to $K&lt;10$.

The Low‐Temperature Thermoelectric Power and Thermal Conductivity of GeTe and of Some GeTe‐MnTe Alloys

AbstractA method of preparation has been developed which allows the growth of large homogeneous single phase ingots of such alloy systems as GeTe‐MnTe, GeTe‐SnTe, etc. … We have measured the thermoelectric power and the thermal conductivity of both GeTe and two alloys of the system Ge1−xMnxTe, in the temperature interval 2.5 to 110 °K, the thermoelectric measurements being extended to 350 °K. Using conventional transport theory both the lattice and the electronic contributions to these effects have been obtained. The temperature dependence of these properties was found to be of similar shape for the GeTe‐MnTe samples as for the pure GeTe, even below the known magnetic transition temperature of these samples. To explain the behaviour of the diffusion term of the thermoelectric power, however, the influence of the incomplete Mn d‐band on the electronic properties of the system must be invoked. The lattice thermoelectric power has the same temperature dependence as the phonon‐drag term in metals. From the analysis of both this component and the lattice thermal conductivity, some informations on the predominant phonon scattering mechanisms have been obtained.

Thermal conductivity of liquid semiconductor thallium-tellurium solutions

Abstract A study has been made of the thermal conductivity K of liquid thallium-tellurium solutions in a composition range where they have semiconductor behaviour. The data are compared with conventional theories for electronic structure and transport, making use of existing information about the electrical properties of these solutions. At a composition which is n-type, but not far from intrinsic, a rapid rise in the experimental value of K with increasing T can be ascribed, at least in part, to ambipolar diffusion. At p-type compositions, the thermal conductivity is found to vary linearly with the product of the electrical conductivity and the absolute temperature over considerable ranges of temperature and concentration, in accordance with the Wiedemann-Franz law. However, the slope differs appreciably from the value for the Wiedemann-Franz ratio derived from conventional transport theory. Some of the possible reasons for the observed behaviour are discussed.

Electronic Transport in Semimetallic Cerium Sulfide

The electrical resistivity $\ensuremath{\rho}$ and Seebeck coefficient $S$ have been measured between 10 and 1000\ifmmode^\circ\else\textdegree\fi{}K in ${\mathrm{Ce}}_{3\ensuremath{-}x}{\mathrm{S}}_{4}$ with values of $x$ ranging from 0 to 0.30. The electrical behavior is semimetallic and can be fitted, in large measure, to the equations of conventional transport theory for $S(T)$ and $\ensuremath{\rho}(T)$. However, examination of the electrostatic effects of the vacancies shows that they introduce large effective charges which are poorly screened. Consequently, there are wide fluctuations in potential in the crystal which cast doubt on a literal interpretation of the theoretical equations; at present, they must be regarded as providing a largely empirical description of the experimental results. There is a relatively large residual resistivity which indicates a very large cross section per vacancy in samples with small values of $x$. This can be accounted for by the abnormally large screening distance. In the temperature range below 100\ifmmode^\circ\else\textdegree\fi{}K, anomalies are observed in $S(T)$ and $\ensuremath{\rho}(T)$ in samples with small vacancy concentrations. Anomalies in $S(T)$ seem to be caused by phonon drag. The cause of the resistivity anomalies is not yet clear; we consider the possibilities that they are caused by local lattice vibrations or by spin scattering by electrons in the $4f$ shell of the cerium ions.

Effect of Reflecting Boundaries on the Transport of Resonance Radiation

The Holstein-Biberman theory of the transport of resonance radiation in gases is generalized to include the case in which the boundary is partially reflecting. The problem is formulated in a manner which follows conventional transport theory somewhat more closely than the earlier treatments of Holstein and Biberman. Using the incoherent scattering approximation, we first write down the two coupled Boltzmann equations describing the mutual transport of excited atoms and resonance photons. The photon equation is then solved, with (diffuse) reflecting boundary conditions, by a slight extension of the interreflection method, and inserting the result into the excited atom equation leads at once to the appropriate generalization of the Holstein-Biberman transport equation. The kernel of this integrodifferential equation, G(r,r′), represents the mean probability that a resonance quantum emitted at r′ is absorbed at r, taking into account the contribution of paths which strike the boundary an arbitrary number of times; G(r,r′) is expressed quite generally in terms of the resolvent kernel of the interreflection equation. The theory is used to calculate the effect of a slightly reflecting boundary on the imprisonment time in a gas discharge.

Invariant imbedding and variational principles in transport theory

A single variational problem is derived which yields the linear equations of conventional transport theory when treated by means of the calculus of variations, and which yields the nonlinear equations of invariant imbedding when approached by means of the functional equation techniques of dynamic programming. The problem considered is a steady-state transport process, involving absorption, fission and scattering, taking place in a one-dimensional rod. For the sake of simplicity, the rod is homogeneous and isotropic. At the conceptual level, the results provide a unified approach to the treatment of internal and external fluxes along classical and modern lines. At the analytic and computational levels, the results enable the calculation of various inequalities for the reflected fluxes and the application of the Rayleigh-Ritz techniques to the determination of fluxes and critical lengths. (N.W.R.)

Spin-Orbit Coupling and the Extraordinary Hall Effect

In general terms a qualitative account of the extraordinary Hall effect in ferromagnetic media may be based on spin-orbit coupling. The magnitude of the effect produced is, classically, very sensitive to the actual path of the electron, or, quantum-mechanically, to some of the detail of its wave function. Here conventional quantum-mechanical transport theory is used in some quantitative detail to evaluate the actual magnitude of the extraordinary Hall coefficient Rs. The initial states of electrons after collisions with the lattice are assumed to be a superposition of Bloch functions with random phases. Bloch functions for the s- and d-bands of nickel are chosen in sufficient detail to make possible a numerical calculation. It is found that Rs/ρ2, where ρ is the resistivity, increases very slightly for the pure metal with increasing temperature. At 290°K, Rs is found to be slightly less than -2 × 10-12 ohm cm gauss-1.