Kubo Theory Research Articles

Valence and conduction band molecular-orbital topologies and the optical and electrical properties of gallium arsenide and silicon

The local electronic valence and conduction band densities of states and corresponding chemical bonding characters of GaAs and silicon are compared on the basis of SCF-X α scattered-wave cluster molecular-orbital computations. The computed densities of states are in good agreement with available spectroscopic data. The calculations indicate that the tops of the valence bands of both materials correspond primarily to nearest-neighbor pσ- and ϱπ-bonding molecular orbitals. However, the molecular-orbital topologies at the bottoms of the conduction bonds of GaAs and silicon are quite different. For GaAs the conduction band edge corresponds to molecular orbitals which are of localized Ga(s, p)As(s, p) σ- and π-antibonding character between nearest neighbors and of delocalized Ga(s)Ga(s) σ-bonding character between second-nearest neighbors. The molecular-orbital topology at the silicon conduction band edge, on the other hand, is of localized Si(s)Si(p) σ-antibonding character between second-nearest neighbors and of localized Si(d)Si(s, p) σ- and π-bonding character between nearest neighbors. These results, in conjunction with Kubo's theory, explain why the electron mobility of GaAs is much larger than that of silicon and why the hole mobilities are similar. This also suggests a “real space” understanding of electron and hole mobility may be related to the localized nature of certain molecular orbitals.

Stochastic Liouville Equation for Weakly Driven System. II: Application to a Quantal Spin

The time-convolutionless stochastic equation derived in a previous paper is applied to a quantal spin interacting weakly with its surroundings under decoupled initial condition, up to second order in powers of random perturbations. The transverse and longitudinal susceptibilities are derived from the stationary solutions. The transverse and longitudinal spectra are displayed and are compared with the ones derived from the time-convolution equation in the conventional Markoffian approximation, and with the ones by the method of second-order spin relaxation theory and Kubo's theory. It is shown that the line shape of transverse spectrum is enhanced over the Lorentzian at the resonance region, due to the effects of interference of the external driving field and the random interaction.

Single‐Ion Anisotropy and ESR Linewidth in Quasi‐Low‐Dimensional Heisenberg Magnets

AbstractOn the basis of the Kubo and Tomita theory the influence of single‐ion anisotropy on the ESR linewidth in a low‐dimensional Heisenberg magnet is estimated in the high‐temperature limit. Two spin correlation functions in r‐representation are used. The variation of the linewidth Δ𝒦(K) depends on the sign of anisotropy K, the value of spin S, and the magnetic dimension of system d. The linewidth can decrease only to a limited value, equal at S 𝒦, Δ𝒦(K) = Δ𝒦(0) · (2/5)(d+1)/3.

Profiles of the ν s(XH) mode in IR spectra of hydrogen-bonded complexes. Theoretical approach

A model describing the phase relaxation of the ν s(XH) mode of a hydrogen-bonded complex XH … Y in an inert solvent is presented. This model includes the coupling between the ν s(XH) and ν s(XH … Y) modes as well as dipole-dipole interactions between the molecular dipole moment μ directed along the (XH) axis and the dipole moment Δμ generated by hydrogen-bond formation. The model assumes that the dipolar coupling can be treated as a stochastic process due to thermal motions of molecules engaged in H-bonding. It is shown that both mechanisms are equally important. The model can be analyzed in terms of Kubo's theory.

On the characterization of the stationary state of a class of dynamical semigroups

We consider a representation of the entropy production for a completely positive, trace-preserving dynamical semigroup satisfying detailed balance with respect to its faithful stationary state denned on aW*-algebraℬ(ℋ): it is expressed as a positive Hermitian form onℬ(ℋ), which is analogous to the quantum correlation functions used in the Kubo theory. By considering this Hermitian form as a variation function of a vector inℬ(ℋ), an exact characterization of the stationary states of semigroups in a certain class is obtained. On this basis, the problem of characterizing the stationary states discussed by Spohn and Lebowitz for manyreservoir open systems is solved without the restriction to situations near thermal equilibrium.

Vibrational relaxation of hydrogen-bonded species in solution. I. Theory

We present a simple model to describe the phase relaxation of the ν s(XH) mode of a hydrogen-bonded species XH⋯Y in solution in an inert solvent. The major assumptions made are that the ν σ(XH⋯Y) stretching mode of the hydrogen bond can be represented by the Ornstein—Uhlenbeck stochastic process, and that the ν s(XH) phase coherence is lost because of the coupling between the two vibrational modes. The model can be analysed in terms of Kubo's theory of a randomly modulated oscillator. An expression is derived for the transition dipole moment autocorrelation function which characterises the line shape of the mid infrared ν s(XH) absorption band, and various limiting cases of the formula are discussed. It is further shown that vibrational relaxation may be expected to influence the far infrared ν σ(XH⋯Y) bandshape and the one-phonon neutron inelastic scattering cross section, and that the parameters required to characterise all three types of spectra are closely related. Experimental tests of the theory are reported in the following paper.

Spin wave excitations in amorphous itinerant ferromagnets II. Dynamical susceptibility

The Izuyama, Kim, and Kubo theory of dynamical susceptibility in crystalline itinerant ferromagnets is extended to the amorphous itinerant ferromagnets. Effects of amorphousness in spin wave spectrum, given by the poles of the susceptibility, is discussed.

Statistical-Mechanical Theory of Random Frequency Modulations and Generalized Brownian Motions

It is shown that the Heisenberg equation of motion can be transformed exactly into the reduced form dA(t)/dt= [i.Q- S:</!(s)ds] ·A(t) +g(t), where A(t) is an arbitrary column vector of dynamical variables, .Q IS a frequency matrix determining collective oscillations, and g (t) IS a fluctuating force. <P (t) differs from the memory function and is related to the time-correlation function of g(t) by </J(t) = (g(t), g*(O)) ·(A (t), A* (0)) _,_ This equation of motion is shown to give a statistical-mechanical theory of random frequency modulations, including Kubo's theory of the stochastic Liouville equation, and to lead to a new theory of generalized Brownian motions which complements a memory-function theory given by Mori.

Theory of exchange narrowing in one and two dimensions

We show that the theoretical approximations in the Kubo and Tomita theory of electron paramagnetic resonance are such as to make its application to one- and two-dimensional systems dubious. A self-consistent theory based upon the independent-mode approximation, that does not have the defects of the Kubo and Tomita theory, is described, and line shapes are calculated for the strongly narrowed limit. Both theories are in significant disagreement with the experimental results on tetramethyl manganese chloride.

Spin wave excitations in amorphous itinerant ferromagnet

The Izuyama, Kim and Kubo theory of spin waves in crystalline itinerant ferromagnets is extended to the amorphous itinerant ferromagnetic materials. The qualitative features of this theory are similar to that of the generalized Landau-Lifshitz theory discussed by Henderson and de Graaf.

The g‐factor of the conduction electrons in Bi2Se3

AbstractThe oscillatory part of the magneto‐resistance (Shubnikov‐de Haas effect) was investigated on n‐Bi2Se3 single crystals with carrier concentrations between 5 × 1017 and 1 × 1018cm−3 in the temperature range from 1.6 to 4.2 K in pulsed magnetic fields up to 35 T (350 kG) magnetic induction. Most experiments were performed for the magnetic induction B parallel to the trigonal axis c of the crystals, where the smallest cyclotron mass mc and a parabolic ε(κ) relation is present. No spin‐splitting of the Shubnikov‐de Haas resistivity extrema is observed at high quantum numbers. At low quantum numbers the influence of spin‐splitting causes a shift of the resistivity extrema and an additional maximum, which should not be present for M = gmc/(2m0) ≪ 1. A calculation of the magnetoresistance in quantizing magnetic fields based on Kubo's theory for a field‐independent carrier concentration enables the determination of the expression M = gmc/(2m0) = 2 with ±5% uncertainty for B‖ c. The absolute value of the g‐factor is evaluated hence as (32 ± 3) for B‖ c. For arbitrary crystal orientation in the magnetic field M may be assumed to be constant within the error margin, the g‐factor being consequently smaller for B ⟂ c: g ⟂ = (23 ± 3).

Equations for the Theory of Response and Transport in Statistical Mechanics

Taking the viewpoint that a complete solution of the Liouville equation contains much more information than is necessary in response theory (wherein one is concerned with a particular response), projection techniques are employed to introduce a shortcut in the problem. The treatment supplements and completes the formalism initiated in a previous publication and yields a general equation which is more relevant to response theory in the usual sense. The most natural application is to the theory of transport. The general transport equation is analyzed in the context of electrical conduction. Meaningful similarities to the simple classical Drude model of conduction are exhibited; the relationship of this exact formalism to the Kubo theory of linear response is shown, an exact solution for a step-function stimulus and approximation procedures which go beyond the linear regime are presented, and an expression for the electrical resistivity is explicitly calculated thus making contact with an earlier paper by Kenkre and Dresden. The response equation is also analyzed in the context of electric and magnetic polarization.

Electron spin resonance spectrum of a new anthraquinone aglycone (Nodosin).

The Electron Spin Resonance study of 1,3,5,8-tetrahydroxy 2,7-dimethyl anthraquinone known as Nodosin has been carried out at room temperature and at X-band. The compound gives a norrow ESR line. The measurements of lineshape, linewidth and g-value have been made. A theoretical computation of dipolar width as well as narrowing due to exchange has been carried out with the help of KUBO and TOMITA theory and finally a value for J, the exchange coupling parameter has been estimated. On the basis of these measurements Nodosin has been proposed to adopt an intermediate structure.

Generalized Mobility Theory of Electron Transfer Reactions in Solution

This paper reports a generalized theory of electron transfer reactions in solution; such reactions are typically termed oxidation-reduction reactions. The theory involves the use of the Kubo linear response analysis to examine the transport processes in an electrolyte system. In particular, we examine a system in which electron transfer between ions in the solution is possible. The substance of the theory is contained in an expression for the electron transfer rate constant. This rate constant is general, valid for all degrees of coupling between the electrons, ions, and surrounding dielectric medium. An expression of this type has not been reported previously. In deriving this general rate constant we find that there are new, hitherto unexpected processes which contribute to the electron transfer rate. Such processes are evident when, through the use of the Kubo theory, we examine the probability of electron transfer correlated with a simultaneous ionic diffusion step and a polarization displacement in the dielectric continuum. The degree to which these various transport processes contribute to the electron transfer rate is examined by means of a calculation on a model system. The system consists of two ionic species, one an electron donor and the other an acceptor, immersed in a dielectric continuum. The transfer electron is considered harmonically bound to the donor ion. The analysis is carried out without recourse to the Born-Oppenheimer separation approximation. We ignore the interaction between the transfer electron and the surrounding ions, but include a brief discussion of ways of handling this problem in the Appendix. This analysis of a model system points the way to future more detailed analyses of the problem.

ESR line width and exchange narrowing in recrystallized samples of hydrazyl and carbazyl

A theoretical computation of the dipolar width ω210 and the exchange frequency Ω20 in powdered samples of DPPH and picryl-n-amino carbazyl (PAC) recrystallized from different solvents has been carried out. Kubo and Tomita theory of ESR line widths has been utilized. The broadening of ESR lines in the recrystallized samples of free radicals has been interpreted in terms of the changes produced in exchange narrowing caused by the molecular-addition complex formation of the free radicals with the solvent molecules. A suitable value of asymptotic Curie temperature has been determined for each sample.

Integrodifferential Equation for Response Theory

The problem of response theory in statistical mechanics involves the determination of the density matrix $\ensuremath{\rho}$ from the Liouville equation and the subsequent computation of the response $r$ from this $\ensuremath{\rho}$. Projection techniques are applied to avoid the entire complicated problem of the full dynamics of $\ensuremath{\rho}$ and to select only that part of $\ensuremath{\rho}$ which is relevant to the response $r$. The procedure replaces an inhomogeneous equation by a linear homogeneous integrodifferential equation for response theory. This is a very general equation which can be analyzed in different ways to yield a variety of results. It is shown that the Kubo theory of linear response emerges as the lowest-order approximation. The general equation is solved without approximations for a step-function stimulus, and it is discussed in the context of the steady state.

Calculation of the E.S.R. line shape in highly viscous media

A method for calculation of the E.S.R. line shape is developed for the case when the Kubo and Tomita theory is invalid. The method is based on the equivalence of functional integration over trajectories and solution of the Fokker-Planck differential equation with some potential. The adiabatic approximation is used in establishing the form of the solution.

Conductivity of a Plasma in a Steady Magnetic Field

We derive expressions for the complex conductivity tensor of a homogeneous classical plasma in an external uniform magnetic field, in terms of electric field correlations, using the Kubo theory of transport phenomena. The main aim is to bring out explicitly the magnetic field dependence of the conductivity tensor. Exact relations between the conductivity tensor in the presence of the magnetic field and the same tensor in the absence of the magnetic field have been obtained.

A fluctuation-dissipation theorem approach to mechanical relaxation in solid polymers

Abstract Kubo theory formalism has been used to obtain expressions for shear and dilatational stress relaxation functions in terms of statistical mechanical time-dependent correlation functions. This is equivalent to obtaining expressions for the complex modulus or the complex viscosity for all frequencies. These results provide a basis for calculating the macroscopic consequences of molecular models presently used to provide qualitative understanding of relaxation peaks for solid polymers. The shear and dilatational stress relaxation functions are quite different formally. For a particularly simple model it will be shown that the former is related to the frequency distribution of the kinetic energy and is also closely related to the dielectric relaxation function. The familiar results of the Rouse model are recovered in the results but no friction constant need be assumed in the present approach.

A theory of dielectric relaxation in polar liquids

Two types of methods have been previously used to relate the macroscopic decay of polarization in a dielectric specimen to a microscopic decay function defined in terms of molecular motions. The first is based on Kubo's theory of non-equilibrium statistical mechanics combined with Kirkwood's theory. The second is based on the use of Debye's rotational diffusion equation. Glarum has made a calculation of the first type, which predicts an exponential macroscopic decay in the case of a liquid assumed to have particularly simple properties. Scaife and the authors have made a calculation of the second type, which for the same liquid agree with each other in predicting a non-exponential macroscopic decay, but disagree with Glarum's calculations. In this paper it is shown that, if Glarum's calculation is modified at one point, the results then agree with the calculations of the second type made by Scaife and the authors. The modification consists in the use, at a certain point, of a macroscopic decay function which we feel to be more justified physically than that used by Glarum.