We review the methods which have been developed over the past several years to determine the behavior of solids whose properties vary randomly at the microscopic level, with principal attention to systems having composition variation on a well-defined structure (random "alloys"). We begin with a survey of the various elementary excitations and put the dynamics of electrons, phonons, magnons, and excitons into one common descriptive Hamiltonian; we then review the use of double-time thermodynamic Green's functions to determine the experimental properties of systems. Next we discuss these aspects of the problem which derive from the statistical specification of the microscopic parameters; we examine what information can and cannot be obtained from averaged Green's functions. The central portion of the review concerns methods for calculating the averaged Green's function to successively better approximation, including various self-consistent methods, and higher-order cluster effects. The last part of the review presents a comparison of theory with the experimental results of a variety of properties---optical, electronic, magnetic, and neutron scattering. An epilogue calls attention to the similarity between these problems and those of other fields where random material heterogeneity has played an essential role.

The theory and properties of randomly disordered crystals and related physical systems

Characteristics of level broadening and transverse conductivity of a two-dimensional electron system under extremely strong fields have been theoretically investigated in the simplest approximation without the difficulty of divergence, i e. in the so-called damping theoretical one. Those of various cases of short- and long-ranged scatterers have been obtained. To see the dependence on the range explicitly, numerical calculation has been performed for the system with scatterers with the Gaussian potential. Especially in case of short-ranged ones the peak value of the transverse conductivity has been shown to be \((N+1/2)e^{2}/\pi^{2}\hbar\) which depends only on the natural constants and the Landau level index. It has been argued from general point of view that this fact is approximately true without reference to kinds of approximations. Such characteristic of the conductivity was confirmed experimentally, but there still remain some problems as to the absolute value of the level width.

Theory of Quantum Transport in a Two-Dimensional Electron System under Magnetic Fields. I. Characteristics of Level Broadening and Transport under Strong Fields

A six-band k⋅p model has been used to study the mobility of holes in Si inversion layers for different crystal orientations, for both compressive or tensile strain applied to the channel, and for a varying thickness of the Si layer. Scattering assisted by phonons and surface roughness has been accounted for, also comparing a full anisotropic model to an approximated isotropic treatment of the matrix elements. Satisfactory qualitative (and in several cases also quantitative) agreement is found between experimental data and theoretical results for the density and temperature dependence of the mobility for (001) surfaces, as well as for the dependence of the mobility on surface orientation [for the (011) and (111) surfaces]. Both compressive and tensile strain are found to enhance the mobility, while confinement effects result in a reduced hole mobility for a Si thickness ranging from 30 to 3 nm.

Six-band k⋅p calculation of the hole mobility in silicon inversion layers: Dependence on surface orientation, strain, and silicon thickness

The broadening of Landau levels and the transport quantities, such as the transverse and Hall conductivity and cyclotron resonance linewidth, are calculated in an inversion layer on the (100) surface of p -type Si at zero temperature. Main scattering mechanisms are assumed to be charged impurity scattering and surface roughness scattering. A new expression for the surface roughness scattering is obtained. The self-consistent Born approximation is employed for the effect of scattering, and the random phase approximation for the screening. Because of the singular density of states the screening depends on the position of the Fermi level and becomes weak when it lies at the tail region of each Landau level. Scattering potentials become strong and of slowly varying type in this case. Except such a special case, overall features agree with the results obtained for short-ranged scatterers.

Screening Effect and Quantum Transport in a Silicon Inversion Layer in Strong Magnetic Fields

For more than 40 years it was thought that polaron- and exciton-phonon systems exhibited unexpected localization properties. Particular attention was paid to the so-called phonon-induced self-trapping transition, which, it was believed, should manifest itself as a point of nonanalyticity in the ground-state energy as a function of the electron-phonon coupling parameter. It will be demonstrated for a large class of (generalized Fr\"ohlich) models that no such transition exists. The dimensionality of space has no qualitative influence; insofar, an application of the authors' results to problems in lower dimensions (e.g., polarons in quantum wells) is straightforward. The same holds true if homogeneous external fields are involved; for example, a discontinuous mass stripping for magnetopolarons can be excluded. On the other hand, a phase-transition-like behavior will be found, if a polaron or exciton is exposed to a short-range potential, allowing a so-called pinning transition. The authors emphasize, however, that even in this case the transition is only modified, and not induced, by phonons.

Analytical properties of polaron systems or: Do polaronic phase transitions exist or not?

Detailed theoretical analysis of non-Ohmic transport of electrons on the liquid 4 He surface is given. The correct form of the electron-ripplon scattering as well as the electron- 4 He gas scattering is taken into account. A characteristic electric field at which electron mobility deviates from the Ohmic value is estimated as a function of temperature and the holding field on the basis of the electron effective temperature approximation.

Warm Electrons on the Liquid 4He Surface

In order to calculate the effective mass and free energy of a polaron at finite temperatures, Feynman's path-integral formalism is generalized in the following two points: (a) the most general quadratic form is used to fully optimize the partition function, and (b) the new definition of the effective mass is introduced such that the effective mass is determined from the acceleration rate against the fictitious driving force which is incorporated in the original Lagrangian. The use of (a) leads to a non-linear integral equation which determines the best trial form. This non-linear integral equation is solved analytically for the Frohlich polaron model to obtain the explicit expressions for the effective mass and free energy at finite temperatures.

Theory of a Polaron at Finite Temperatures

By using the Green function method for the random lattice, the susceptibility of the impurity band in weak magnetic fields is investigated in a simplified model in which the impurity potential is attractive and of short range and the electron-electron correlation can be negligible. Simple analytical expressions for the susceptibility are obtained and are computed numerically as a function of the concentration of impurities c . When c / c 0 ( c 0 is the critical concentration at which the impurity band touches the conduction band) is very small so that the impurity band width is smaller than the binding energy of the impurity potential, the orbital susceptibility in the impurity band region is simply of the type of the atomic diamagnetism, while it is of the type of the ordinary Landau diamagnetism in the conduction band region. The result for the case c / c 0 ≫1 explains the observed discrepancy from the existing Landau theory.

Theory of Impurity Bands in Magnetic Fields. I. Magnetic Susceptibility

The Hall coefficient and the magnetoresistance in a weak magnetic field are investigated for the impurity conduction processes based on a simplified model where the impurity potential is short-ranged. The formation of the impurity band and the modification of the intrinsic band are treated on the same footing by the Green function method. It is shown that the sign of the Hall coefficient is always electronic for n -type semiconductors irrespective of the degree of compensation. Two origins of the magneto-resistance, the Zeeman spin splitting and the Landau orbital motion, are separately calculated and both longitudinal and transverse magnetoresistances are shown to be of the same order of magnitude in the impurity band region. Numerical calculations of these transport coefficients are carried out as functions of the impurity concentration and of the degree of compensation. Intuitive discussion is made on the intrinsic nature of the impurity conduction processes. Further, the relation between the Hall coef...

Theory of Impurity Bands in Magnetic Fields. II. Transport Properties

A general expression for the galvanomagnetic coefficients of hot electrons in an anisotropic medium is derived on the basis of the Boltzmann transport equation. The expression is applied to a nondegenerate piezoelectric semiconductor to understand the observed anisotropy and the field dependence of the Hall mobility of cadmium sulfide. Two mechanisms, piezoelectric and deformation potential scatterings, are considered, and a distribution function is derived by solving the Boltzmann equation directly. Satisfactory agreement between theory and experiment is obtained by taking the electron effective mass 0.17 m 0 where m 0 is the bare electron mass and the deformation potential constant \(\varXi{=}16{\pm}2\) eV. The anisotropic effect at low temperatures is concluded to be due to the anisotropy of piezoelectric tensors which characterizes the scattering. Effects due to other scattering mechanisms are discussed.

Motohiko Saitoh

Papers

Warm Electrons on the Liquid 4He Surface

Theory of a Polaron at Finite Temperatures

Theory of Impurity Bands in Magnetic Fields. I. Magnetic Susceptibility

Theory of Impurity Bands in Magnetic Fields. II. Transport Properties

Nonlinear Galvanomagnetic Phenomena in Cadmium Sulfide