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The theory of Bloch electron dynamics for carriers in homogeneous electric and magnetic fields of arbitrary time dependence is developed in the framework of the Liouville equation. The Wigner distribution function (WDF) is determined from the single particle density matrix in the ballistic regime, i.e., collision effects are excluded. The single particle transport equation is established with the electric field described in the vector potential gauge, and the magnetic field is treated in the symmetric gauge. The general approach is to employ the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including multiband Zener tunneling, is treated exactly. In the formulation of the WDF, we transform to a new set of variables so that the final WDF is gauge invariant and is expressed explicitly in terms of the position, kinetic momentum, and time. The methodology for developing the WDF is illustrated by deriving the exact WDF equation for free electrons in homogeneous electric and magnetic fields. The methodology is then extended to the case of electrons described by an effective Hamiltonian corresponding to an arbitrary energy band function. In treating the problem of Bloch electrons in a periodic potential, the methodology for deriving the WDF reveals a multiband character due to the inherent nature of the Bloch states. In examining the single-band WDF, it is found that the collisionless WDF equation matches the equivalent Boltzmann transport equation to first order in the magnetic field. These results are necessarily extended to second order in the magnetic field by employing a unitary transformation that diagonalizes the Hamiltonian using the ABR to second order. The work includes a discussion of the multiband WDF transport analysis and the identification of the combined Zener-magnetic field induced tunneling., Comment: 23 pages

We have investigated the accuracy of the local-spin-density approximation with orbital-density-dependent self-interaction correction (LSDSIC) as proposed by Perdew and Zunger within a Kohn-Sham approach in which electrons with a given spin projection all move in a single optimized effective potential (OEP). We have also studied the accuracy of the Krieger-Li-Iafrate (KLI) approximation to the OEP for the same energy functional in order to assess its applicability to systems in which the integral equation for the OEP cannot be reduced to a one-dimensional problem, e.g., molecules. Self-consistent Kohn-Sham LSDSIC calculations have been performed for atoms with atomic number Z=1--20 in the exchange-only case for the total energy, the highest-occupied orbital energy ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{m}}$, and the expectation value of ${\mathit{r}}^{2}$. In addition, the structure of the resulting exchange potential is examined and compared with the exact exchange-only density-functional theory (OEP method with Hartree-Fock exchange-energy functional) results. Furthermore, we display ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{m}}$, the ionization potential I, and the electron affinity A when both exchange and correlation energy effects are included. Finally, we also consider the results of evaluating the LSDSIC energy functional by employing the exact (in the central-field approximation) single particle orbitals as proposed by Harrison. We find that the LSDSIC energy functional generally leads to calculated values that are superior to those provided by the LSD approximation and that the KLI approximation yields results in excellent agreement with the corresponding exact OEP results for this energy functional. In particular, quantities strongly related to the behavior of the valence electrons are nearly identical in both the OEP and KLI calculations, i.e., the difference between the 〈${\mathit{r}}^{2}$〉 and ${\mathrm{\ensuremath{\epsilon}}}_{\mathit{m}}$ is less than 0.2% on average, while the difference between the calculated I is less than 0.2 millihartree on average with the corresponding difference of only 0.1 millihartree for A. \textcopyright{} 1996 The American Physical Society.

The $g$ factors and mean lifetimes of several short-lived low-lying states in ${}_{30}^{70}$${\mathrm{Zn}}_{40}$ have been measured using the techniques of projectile Coulomb excitation in inverse kinematics combined with transient magnetic fields and the Doppler-shift attenuation method. The present results have been interpreted within the framework of large-scale shell-model calculations that include the ${g}_{9/2}$ orbital.

The optimized-effective-potential (OEP) method and a method developed recently by Krieger, Li, and Iafrate (KLI) are applied to the band-structure calculations of noble-gas and alkali halide solids employing the self-interaction-corrected (SIC) local-spin-density (LSD) approximation for the exchange-correlation energy functional. The resulting band gaps from both calculations are found to be in fair agreement with the experimental values. The discrepancies are typically within a few percent with results that are nearly the same as those of previously published orbital-dependent multipotential SIC calculations, whereas the LSD results underestimate the band gaps by as much as 40%. As in the LSD---and it is believed to be the case even for the exact Kohn-Sham potential---both the OEP and KLI predict valence-band widths which are narrower than those of experiment. In all cases, the KLI method yields essentially the same results as the OEP.

A theory for the spontaneous emission of radiation for a Bloch electron traversing a single energy band under the influence of a constant external electric field is presented. The constant external electric field is described in the vector potential gauge. The quantum radiation field is described by the free space quantized electromagnetic field in the Coulomb gauge. The instantaneous eigenstates of the Bloch Hamiltonian are introduced as basis states to analyze the Bloch dynamics to all orders in the constant external electric field. The radiation field described by the quantized electromagnetic field provides the vacuum fluctuations that are responsible for the spontaneous emission of radiation of the single electron from the upper regions of the energy band. It is shown that the spontaneous emission occurs with frequencies equal to integral multiples of the Bloch frequency without any ad hoc assumptions made concerning the existence of Wannier-Stark ladder levels. An explicit expression for the spontaneous emission transition probability is derived to first order in the quantized radiation field; results show the explicit dependence upon the electron energy band structure, photon polarization, and the directionality of the radiation output. As an illustration, spontaneous emission probabilities are developed and illustrated for nearest-neighbor tight-binding and more realistic superlattice band structure models. For the GaAs-based superlattices, the power radiated into free space from spontaneous emission due to Bloch oscillations in the terahertz frequency range is estimated to be about one-tenth of a microwatt.

Coupled-cluster calculations starting from exchange-only local-density approximation (XLDA), Krieger-Li-Iafrate (KLI), and Kohn-Sham (KS) wave functions are compared with those using the Hartree-Fock (HF) determinant as areference. The total energies are found to be close, the difference being maximally 2 mhartree in the systems studied (the first terms in the He, Be, Ne, Mg. Ar isoelectronic series). The convergence is, however, sensitive to the choice of the reference: KLI and KS converge, in general, faster than HF in spite of being a worse approximation in the first two orders of perturbation theory. The improvement of convergence due to the use of the KLI or KS references is more pronounced in the systems showing near degeneracy, such as in the Be series. For XLDA, the convergence properties are either comparable to those of KLJ or oscillatory, depending on the system. In a second part, the numerical results are analyzed (in the HF and KLI cases) by using first-order developments with respect to nuclear charge Z at large Z.

The theory of a Bloch electron moving in the presence of a homogeneous electric field is reviewed and objections to the conventional derivations are discussed. A new derivation of the time development of a Bloch electron moving in a homogeneous, but time-dependent, electric field is presented using a vector potential to describe the field rather than the usual scalar potential. This new treatment avoids all the basic assumptions of the conventional derivations and demonstrates that a Bloch electron will oscillate in a single band with the Bloch period if a homogeneous electric field is abruptly turned on, with a tunneling probability into other bands given by the conventional expression. It is also shown that the calculated optical absorption will have the same ladderlike structure that would be obtained if Wannier-Stark quantized energy levels are assumed, although the present calculation makes no such assumption. The previous objections to the existence of Bloch oscillations for electrons in a perfect periodic potential are examined and found to be irrelevant provided the tunneling probability per Bloch oscillation period is much less than one, a condition that is generally satisfied for typical elemental and compound semiconductors for electric fields smaller than ${10}^{6}$ V/cm.

A new method of constructing approximate Kohn-Sham exchange-correlation potentials, ${v}_{\mathrm{xc}}$, from the orbital-dependent potentials ${\mathit{v}}_{\mathrm{xc}}^{(\mathit{i})}$=\ensuremath{\delta}${\mathit{E}}_{\mathrm{xc}}$[{${\mathit{n}}_{\mathit{i}}$}]/\ensuremath{\delta}${\mathit{n}}_{\mathit{i}}$ is proposed in which ${v}_{\mathrm{xc}}$ is obtained from integrating the weighted average of electric (WAE) field produced by the ${v}_{\mathrm{xc}{}^{(i)}}$. Application of the WAE method to atoms with closed subshells employing exchange-only local-spin-density self-interaction correction and gradient-expansion-approximation self-interaction correction energy functionals yields ${v}_{x}$ and n(r) which closely approximate the OPM potentials and Hartree-Fock electron densities, respectively.

A recently developed variational formalism for the determination of the reduced single-particle density matrix, correct to second order, is applied to the ground state of the helium isoelectronic sequence. For a Slater-determinant-- type trial wave function the method requires the initial determination of either the charge density or equivalently its cosine Fourier transform for spherically symmetric systems. The trial wave function employed is a one-parameter Hartree product of hydrogenic functions and use is made of the highly accurate analytic expressions derived elsewhere for the Fourier transform of the charge density for the helium sequence. Analytic expressions for the single-particle density matrix are obtained and the internal self-consistency of the technique with regard to the Kato cusp condition is discussed. These expressions are then employed to obtain closed form analytic expressions for the momentum density valid for the entire isoelectronic sequence. These results are subsequently employed to obtain expressions for the expectation values of the operators p$sup 4$, p$sup 2$, vertical-barpvertical-bar, and vertical-barpvertical-bar$sup -1$ and the Compton profile in the impulse approximation. Analytic Hartree-Fock calculations for these properties are also performed, and the results of the variational calculation are compared with these results and those of many-parameter correlated wave-function calculations wherevermore » possible. It is observed that the results of the single-parameter variational calculation for helium are highly accurate and improve further for each heavier element of the isoelectronic sequence.« less

A novel formalism for treating Bloch electron dynamics and quantum transport in inhomogeneous electric fields of arbitrary strength and time dependence is presented. In this formalism, the electric field is described through the use of the vector potential. This choice of gauge leads to a natural set of basis functions for describing Bloch electron dynamics; in addition, a basis set of localized, electric field-dependent Wannier functions are established and utilized to derive a quantum "Boltzmann equation" which includes explicit band-mixing transients such as effective mass dressing and Zener tunneling. The applications of this formalism to quantum transport in spatially localized inhomogeneous electric fields such as occur in problems involving tunneling through "band-enginereed" tunneling barriers and impurity scattering are discussed.

A variational iteration technique is developed for calculating the distribution function and electrical resistivity for semiconductors with anisotropic band structure. When only a single iteration is performed the result for the resistivity is identical to that obtained by an application of Kohler's principle with a relaxation-time approximation and the calculated distribution function includes anisotropic corrections to the zeroth-order energy-dependent relaxation time. The accuracy of the technique is demonstrated by comparing the results of a first-order calculation with the exact solution of the Boltzmann equation for the transverse resistivity and the distribution function for Brooks-Herring scattering by current carriers moving on an ellipsoidal constant-energy surface in the limit of infinite-mass anisotropy. The technique is applied to the calculation of the longitudinal resistivity and Hall $r$ coefficient of saturation-stressed degenerately doped $n$-type Ge at $T=0\ifmmode^\circ\else\textdegree\fi{}$K. We find that $r$ is within 5% of unity, thus justifying Katz's determination of the number of carriers from his Hall data but the theoretical upper bound to the resistivity is found to significantly underestimate the experimental results. However, when the calculation is approximately corrected for the known inaccuracies of the Born approximation together with contributions from multiple scattering and dressing effects, the theory is found to be in good agreement with experiment.

A variational principle for the reflection coefficient is derived for elastic scattering from three-dimensional crystals. The crystal is assumed to possess perfect two-dimensional periodicity in the plane parallel to the crystal surface, but arbitrary variations of potential and lattice spacing in the perpendicular direction. As a result, this theory is valid for crystals with or without impurity layers. With linear trial functions, the reflection coefficient is uniquely given by the ratio of two determinants without any additional calculation of the wave field inside the crystal. Another variational principle is derived for the transmission coefficient of a plane wave transmitted through a three-dimensional crystal with a finite width. A modified Born approximation, which leads to the Born approximations, is obtained by inserting the incident and the adjoint plane waves in the variational equation without applying the variational principle. A general modified Born approximation is also obtained. For crystals having triperiodicity, the employment of Bloch waves as trial functions determines the reflection coefficient in terms of integrals over known functions and the perpendicular components of the wave numbers of the Bloch waves.

The formalism developed by the authors for the variational determination of the expectation value of single-particle operators $W=\ensuremath{\Sigma}{i}^{}W({\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i})$ via Delves's principle in the Hartree approximation is extended here to the Hartree-Fock approximation by employing a Slater-determinant-type trial wave function and an appropriately antisymmetrized auxiliary function. A set of coupled integral-differential equations for the components of the auxiliary function are obtained by a subsidiary variational minimization of a functional involving the trial wave function, the auxiliary function, the Hamiltonian, and the operator $W$. Owing to the antisymmetric nature of both the trial and auxiliary functions, an exchange term involving the specific single-particle operator in question appears in these equations. Decoupling approximations are discussed and the equations solved exactly for single-particle operators that depend on the radial distance only. Employing a single-parameter appropriately antisymmetrized product of the $1S$ and $2S$ hydrogenic functions as the trial wave function, the formalism in this Hartree-Fock approximation is then applied to both the helium $2 ^{3}S$ and $2 ^{1}S$ isoelectronic sequences to obtain analytic expressions for the expectation value of the operators ${r}^{n}$, $n=2,1,\ensuremath{-}1$, and $\ensuremath{\delta}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})$. The results of these calculations are observed to approximate closely the results of a 2300-parameter calculation due to Accad et al. and in the orthohelium case to be also equivalent to those due to Hartree-Fock.

The work of Kohn is generalized to more than one dimension for a finite-band model by a simple algebraic procedure. Some results not previously derived for more than one dimension are obtained. The differences between the analytic properties of the model and those of the complete Hamiltonian are briefly discussed.

By employing Kohler's principle, an upper bound to the electrical resistivity has been calculated for unstressed $n$-type degenerately doped germanium at $T=0$ \ifmmode^\circ\else\textdegree\fi{}K, assuming Brooks-Herring scattering. The observed resistivity, however, is found to be 5-7 times larger than the theoretical estimate. We find that the usual corrections to Brooks-Herring scattering for semiconductors originally discussed by Moore and Ehrenreich for isotropic bands are not sufficient to account for this discrepancy. Instead, we show that the use of dielectric screening instead of that due to Thomas-Fermi leads to a significant decrease in the intervalley contribution to the screening in many-valley semiconductors with highly anisotropic band structure. When this effect is taken into consideration, we find good agreement between theory and experiment for the resistivity of arsenic-doped germanium.

The analytical method developed by the authors for the determination of the expectation value of single-particle operators $W={\ensuremath{\Sigma}}_{i}{W(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}_{i})$ correct to second order is employed in both the first and second decoupling approximations to obtain analytic expressions for the coherent x-ray scattering factor $F(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ for the ground state of the helium isoelectronic sequence valid for all values of momentum transfer. The trial wave function ${\ensuremath{\psi}}_{0T}$ employed is an energy-minimized Hartree product of hydrogenic states. For helium the results for the form factor in the second decoupling approximation are found to be superior to those of the first, and are within 1.2% of the highly accurate values calculated using a 120-parameter configuration-interaction wave function and have an accuracy equivalent to that of an analytical Hartree-Fock treatment. This error is further reduced as the atomic number is increased. In order to demonstrate the internal self-consistency of the technique, we prove that the expectation value of any single-particle operator as obtained by direct use of the analytical method is the same as the expectation value obtained employing the form factor provided that the latter has also been calculated by the analytical method employing the same trial wave function. Finally, we extend our calculations to the infinite-momentum-transfer range and study our results in this limit by employing a cusp condition for the exact ground-state wave function of two-electron atomic systems written in terms of the logarithmic derivative of the electron density at the origin. We observe that in the infinite-momentum-transfer limit our results for helium are in error by 0.57% and that this error is further diminished for each successive element of the isoelectronic sequence. In addition, we note that the cusp condition is exactly satisfied via the formalism of the second decoupling approximation and is independent of the variational parameter employed.

A variational principle for the reflectance is derived for elastic scattering from one-dimensional potentials. Using this principle, we show that the reflection coefficient is given by the ratio of two determinants without any subsidiary calculation of the wave field in the crystal and without any need to perform a matching on the boundary. The results are valid for crystals having variable lattice constants, including the possibility of impurity layers. For scattering from periodic potentials, the results are most conveniently obtained by employing Bloch's theorem with the wave number inside the crystal obtained from evaluating a Hill's determinant. The variational principle is also employed to obtain a modified Born approximation for the reflectance. We also compare the reflectance given by approximate wave functions with the exact reflectance for the Kronig-Penney model, the latter also having been obtained by the variational principle.

Upper and lower bounds to the transverse resistivity due to ionized impurity scattering in saturation-stressed degenerately doped $n$-type germanium (single conduction-band valley occupied at $T=0$ \ifmmode^\circ\else\textdegree\fi{}K) have been calculated for Brooks-Herring scattering. The bounds are obtained by solving exactly the Boltzmann equation including the mass anisotropy for two different scattering rates which either overestimate or underestimate the Brooks-Herring scattering rate. The use of Kohler's variational principle also yields an upper bound to the resistivity, which is found to be approximately 20% higher than the lower bound previously obtained, but is smaller than the experimental results of Katz. The calculation is approximately corrected for the inaccuracies of the Born approximation together with contributions from multiple scattering and dressing effects. The final results are in excellent agreement with experiment.

Recent calculations of the dielectric enhancement of the resistivity of $n$-type degenerately doped germanium at low temperature is extended to silicon. We find, as in the Ge case, that Brooks-Herring scattering significantly underestimates the resistivity but the inclusion of the effects of dielectric screening is sufficient to remove the discrepancy between theory and experiment.

Electron shielding in heavily doped many valley semiconductors, discussing Thomas-Fermi screening and Mott transition