Spherical Constant Energy Surfaces Research Articles

Investigation of Electron Crystallization in a HCP Structure

The ground state energies of non-magnetic, ferromagnetic and antiferromagnetic phases of 3D electron crystals corresponding to hcp structure are computed. In each case, the possibility of the Wigner electrons having cubic or spherical constant energy surface (the region of integration in momentum space) is investigated. The role of correlation energy is suitably taken into account. The range of the low density region favourable for Wigner electron crystallization is found. The structure dependent Wannier functions, which give proper localized representation for the Wigner electrons in the crystal, are employed in the calculation.

EFFECT OF POSITIVE BACKGROUND ON THE FERROMAGNETIC PHASE OF 3D WIGNER ELECTRON CRYSTAL

In the ferromagnetic phase of the 3D electron system, it is assumed that the spins of all the electrons are arranged in a parallel manner. The ferromagnetically ordered electrons are represented by the structure dependent Wannier functions. The ground state energies of the ferromagnetically ordered electron crystals corresponding to sc, bcc, fcc, diamond and perovskite structures are computed with Gaussian type and Yukawa type positive backgrounds. The stability of these structures is analyzed. In each case, the possibility of the Wigner electrons having cubic or spherical constant energy surface (the region of integration in momentum space) is considered for the range of rs values corresponding to low densities. The effect of correlation energy is suitably taken into account. The range of low density region favorable for Wigner crystallization is found to be at 6.0×1019/ cm 3. It is found that the ferromagnetically ordered electrons crystallize into fcc lattice in the low density limit with Yukawa type positive background.

Electrical and thermoelectric properties of p-Ag2Te in the β phase

The conductivity σ, Hall coefficient R, and thermoelectric power α0 of p-Ag2Te were studied in the temperature range of 300–550 K. Inconsistency between the signs of R and α0 was observed at 420–550 K. These results are interpreted within the two-phase model with spherical constant energy surfaces. It is established that the inconsistency between the R and α0 signs is due to the emergence of the scattering mechanisms with the parameters r0ac, r00, and r0d an increase of about 50% in the ratio of the effective electron and hole masses as a result of the transition α → β.

INVESTIGATION OF THE FERROMAGNETIC PHASE OF WIGNER ELECTRON CRYSTAL

The ground state energies of the ferromagnetically ordered electron crystals corresponding to sc, bcc, fcc, diamond and perovskite structures are computed. The stability of these structures is analyzed. In each case, the possibility of the Wigner electrons having cubic or spherical constant energy surface (the region of integration in momentum space) is considered for the range of r s values corresponding to low densities. The role of correlation energy is suitably taken into account. The range of low density region favourable for Wigner crystallization is found to be above r s = 30. It is found that, the ferromagnetically ordered electrons crystallize into fcc lattice in the low density limit. The structure dependent Wannier functions, which give proper localized representation for the Wigner electrons in the crystal, are employed in the calculation.

STUDY OF STRUCTURAL STABILITY OF WIGNER ELECTRON CRYSTAL

The ground state energies of the non-magnetic Wigner electron crystals corresponding to sc, bcc, fcc, diamond and perovskite structures are estimated and it is found that the bcc lattice still remains to be the stable known arrangement for three-dimensional Wigner electron crystal. Perovskite structure is not the stablest as claimed in a previous work, it is preferred only after fcc and sc. The stability is analysed taking different structures and assuming the possibility of the Wigner electrons having cubic or spherical constant energy surface, the region of occupation in momentum space, for a whole range of rs values (rs=20 to 200). The structure dependent Wannier functions, which give a proper localized representation for the Wigner electrons in the crystals are constructed and employed in the present calculation.

Hole Velocity in an Electric Field for Valence Bands Degenerate at k = 0

AbstractIn p‐type semiconductors having spherical constant energy surfaces and parabolic dispersion law the velocity of the hole immersed in an electric field F, as known, is described by the simple formula Ft/mh,1, where t is the time and mh,1 the effective mass of heavy or light holes. The paper shows that inclusion of intervalence tunneling between heavy and light mass subbands leads to a more complicated time dependence of the hole velocity, especially near the point of degeneracy, where the wave vector k = 0. The curves of the time dependence of hole velocity for parameters typical to p‐Ge and for some values of the electric field and initial hole wave vector are presented. The paper also demonstrates how the theory of angular momentum can be adapted to the problems of hole dynamics in valence bands degenerated at k = 0.

Quasiclassical kinetic equation for charge-carrier transport in a semiconductor

Starting with the Uehling–Uhlenbeck quasiclassical generalized Boltzmann equation, a kinetic equation is derived for the distribution function f(ε,t) for charge carriers of energy ε and effective mass m in a semiconductor in the presence of an electric field E. Charge-carrier scattering with optical and acoustic phonons through strain and polar interactions are taken into account. Introducing a Legendre-polynomial expansion of the distribution function and assuming spherical constant energy surfaces, the following equation emerges in the limit of large charge carrier-to-phonon energy: (∂f0/∂t)+(eE/3) ((2ε/m))1/2 ((∂/∂ε) +(1/ε))(f1(ε,0)e−t/τ−T̂eE ((2ε/m))1/2 (∂/∂ε) f0)=(P−aQf0) (∂f0/∂ε) +(F−aGf0) (∂2f0/∂ε2)+Cf0 (1−af0). Collision integrals P,Q,F,G, and C, as well as the collision frequency τ, are evaluated explicitly. The quantum parameter a is 1 in the degenerate domain and 0 in the classical domain. The first two Legendre components of f(ε,t) are written f0 and f1, respectively, and T̂ is a time-dependent operator. Working in steady state and passing to the classical limit it is found that acoustic-strain interactions dominate over all others. Leading terms in this expansion give a new quasiclassical equation for the distribution of charge carriers of a semiconductor in an electric field. Further reducing this equation in the limits of equipartion and mu/p≪1, where p is carrier momentum and u is acoustic phonon velocity, gives a generalized Druyvesteyn equation which incorporates exclusion effects.

Cascade Capture of Non‐Equilibrium Carriers

AbstractThe low‐temperature galvanomagnetic effects in non‐polar semiconductors are investigated under the condition that the non‐equilibrium carriers are captured by shallow attractive traps. The calculations are carried out assuming spherical constant energy surfaces in the momentum space. Numerical calculations are made for n‐Ge at 4.2 K. The possibility of using the results for the determination of some of the capture parameters is discussed.

Nonradiative multiphonon carrier capture by deep neutral traps

AbstractThe galvanomagnetic effects and the Faraday rotation are investigated in the general case for any mechanism of the energy and momentum scattering of hot electrons in the strong and weak magnetic field limits and under the condition when the free carriers are captured in a nonradiative way by deep neutral traps described by a δ‐like attractive potential. For some definte scattering mechanisms the investigations are carried out in the presence of any finite magnetic field. The possibility of using the results for the estimation of some semiconductor parameters and for the determination of the dominant scattering mechanism is discussed. The calculations are carried out assuming spherical constant energy surfaces in the κ space.

Multipole wave functions for photoelectrons in crystals. II. Examples of constant-energy-surface harmonics. Application to thes−dbands of Cu

We construct a complete set of multipole wave functions for an electron in a crystal, appropriate to describe single-center phenomena. The set consists of superpositions of the Bloch waves of given energy $E$, weighted by a set of harmonics which is complete and orthonormal over the constant-energy surface ${E}_{\ensuremath{\mu}}(\stackrel{\ensuremath{\rightarrow}}{k})=E$. We give prescriptions to label these harmonics. Numerical examples are discussed for the $s\ensuremath{-}d$ bands of Cu. The case of a spherical constant-energy surface, whose harmonics are the spherical harmonics, reappears as a degenerate limit.

Models for phase transitions with continuous constant energy surfaces

We have studied models of phase transitions of a type introduced by Brasovskii. These models involve an infinite number of degenerate order parameters, each associated with a finite nonzero wave vector. Brasovskii discussed the case of a spherical constant-energy surface in reciprocal space. We extend the Brasovskii model to different spatial dimensions $d$ and to different continuous constant-energy surfaces of dimension $m$. The ring model, the model discussed in most detail, has a continuous constant-energy surface described by a two-dimensional circle embedded in a $d$ dimensional space. The order of the phase transition to the nonuniform state is shown to depend on the number of independent directions orthogonal to the constant-energy surface in reciprocal space. When there are more than four directions perpendicular to the constant-energy surface ($d>5$ for the ring model), fluctuations are unimportant and the transition is continuous with mean-field exponents. However, when there are less than four directions ($d<5$ for the ring model), the phase transition to a nonuniform state is shown to be first order, rather than second order, as predicted by mean-field theory. We also present the analogous results for the more general models.

On the theory of electron diffusion under a temperature gradient in degenerate semiconductors having non-parabolic energy bands

The theory of electron diffusion under a temperature gradient (Soret effect) in degenerate semiconductors having non-parabolic energy bands and spherical constant-energy surfaces is worked out for the first time. An expression is given for the Soret coefficient which may be determined from the knowledge of the energy dependence of the relaxation time and can be related to the thermoelectric power.

The voigt-type microwave Kerr effect in semiconductors with spherical constant energy surfaces∗

A plane wave analysis is given for the free-carrier, microwave magneto-Kerr effect in semiconductors having spherical constant energy surfaces for the case of the applied static magnetic field perpendicular to the propagation direction (Voigt-type Kerr effect). The analysis is in terms of R v, the complex polarization factor of the reflected wave. The behavior of R v is considered in detail for low magnetic fields. Multiple-carrier conduction and energy-dependent scattering are shown to give rise to major contributions to R v. Computer curves for | R v|vs. μ e B, n p and ω are presented for parameters corresponding to InSb. Approximate expressions for R v, which are valid for low magnetic field, high-loss conditions, are given and compared with curves computed from the more complex expressions. Room temperature data for R v are presented for TE 11 waves in a circular waveguide. The dependence of R v on magnetic flux density is shown for intrinsic InSb. The data are compared with the plane wave predictions. Experimental data for the magneto-Kerr effect are also given for magnetic fields slightly misaligned from the Voigt orientation. An empirical model is introduced which describes these data in terms of data for the Faraday and Voigt orientations of the magnetic field. This model is shown to be of value for alignment of the magnetic field in the Voigt orientation, and for measurement of the Voigt-type Kerr effect in the presence of any small, remaining longitudinal magnetic field component.

The Voigt-type microwave kerr effect in semiconductors with spherical constant energy surfaces

This chapter discusses the theory of linear elastic wave propagation in crystals. Three different linear elastic waves may propagate along any given direction in an anisotropic material. These three waves are usually not pure modes as each wave has particle displacement components both parallel and perpendicular to the wave normal. When one of these components is normally much larger than the other, the wave with a large parallel component is called quasilongitudinal, while the wave with a large perpendicular component is called quasitransverse. In the event that sufficient symmetry prevails such that the direction of propagation is elastically isotropic or if the material is elastically isotropic, then all modes become pure modes. For linear elastic wave propagation in crystals, the direction of the flow of energy per unit time per unit area, the energy-flux vector, does not in general coincide with the wave normal as it does in the isotropic case. An expression for the energy-flux vector can be obtained by considering the time rate of change of the total elastic energy contained in the wave field. Measurement of the linear elastic properties of solid materials is one of the most important applications of ultrasonic techniques. The measurements range from determination of second-order elastic constants, which yield fundamental information about the binding forces among atoms in crystals, to the field of nondestructive testing where macroscopic internal defects are located in structural solids.

Piezoresistance inn-Type CdTe

The elastoresistance coefficients for $n\ensuremath{-}\mathrm{C}\mathrm{d}\mathrm{T}\mathrm{e}$ were determined at 300\ifmmode^\circ\else\textdegree\fi{} and 195\ifmmode^\circ\else\textdegree\fi{}K. The values of the coefficients are small and are comparable to those in $n\ensuremath{-}\mathrm{I}\mathrm{n}\mathrm{S}\mathrm{b}$ and $n\ensuremath{-}\mathrm{G}\mathrm{a}\mathrm{A}\mathrm{s}$. This result is consistent with the hypothesis that the electrical conduction in $n\ensuremath{-}\mathrm{C}\mathrm{d}\mathrm{T}\mathrm{e}$ is due to the electrons in the $k=0$ band characterized by spherical constant energy surface. The data on the resistance and Hall coefficient as a function of hydrostatic pressure (up to 1.2\ifmmode\times\else\texttimes\fi{}${10}^{10}$ dyn/${\mathrm{cm}}^{2}$) indicate additional complications in the conduction band. Our results are inconclusive about this additional mechanism, which is responsible for sharply decreasing the electrical conductivity in $n\ensuremath{-}\mathrm{C}\mathrm{d}\mathrm{T}\mathrm{e}$ at higher pressures.

Magnetoconductivity of Hot Electrons

A theoretical analysis of galvanomagnetic phenomena in an intense electric field is given. The Boltzmann transport equation is solved for the case of acoustic-phonon and impurity scattering and the magnetoconductivity tensor is calculated. The dependence of the Hall and conductivity mobilities on electric and magnetic fields is determined both for spherical constant-energy surfaces and a band structure of the many-valley type. Expressions are derived for both the Hall and conductivity mobilities in the limit of small magnetic fields, where one considers quadratic variations of mobility in the magnetic fields, as well as the case of large magnetic fields. Negative magnetoresistance obtains for the case of spherical constant-energy surfaces and predominant acoustic-phonon scattering, whereas the sign of the mobility variation is shown to depend sensitively on the many-valley band structure and intervalley transition processes.

Analysis of Lattice and Ionized Impurity Scattering inp-Type Germanium

The scattering of holes in $p$-type germanium by acoustical and optical phonons, and by ionized impurities has been studied over a wide range of temperature from 7 to 300\ifmmode^\circ\else\textdegree\fi{}K, and for impurity concentrations from ${10}^{13}$ to ${10}^{17}$ ${\mathrm{cm}}^{\ensuremath{-}3}$. The mobility was analyzed on the basis of the following approximations: (1) The light- and heavy-hole valence bands are parabolic with spherical constant-energy surfaces and effective-mass ratios of 0.043 and 0.35; (2) the relaxation time for lattice scattering is the same for the heavy and light holes; and (3) the ionized impurity scattering is describable by the Brooks-Herring formula with the scattering limited to intraband transitions. The following results were obtained: (1) The expectation was substantiated that the lattice scattering at $T\ensuremath{\le}70\ifmmode^\circ\else\textdegree\fi{}$K can be attributed to acoustical phonon interactions; the mobility ${\ensuremath{\mu}}_{\mathrm{ac}}=3.37\ifmmode\times\else\texttimes\fi{}{10}^{7}{T}^{\ensuremath{-}\frac{3}{2}}$ ${\mathrm{cm}}^{2}$/V-sec was obtained by treating it as the sole adjustable parameter in the range 10 to 70\ifmmode^\circ\else\textdegree\fi{}K; (2) the stronger ${T}^{\ensuremath{-}2.3}$ dependence of mobility in the range 120 to 300\ifmmode^\circ\else\textdegree\fi{}K can be attributed to the advent of optical mode scattering if the ratio of optical to acoustical mode coupling constants ${(\frac{{\mathcal{E}}_{\mathrm{op}}}{{\mathcal{E}}_{\mathrm{ac}}})}^{2}$ is chosen to be 3.8; (3) the Brooks-Herring formula describes the ionized impurity scattering very well from 30 to 300\ifmmode^\circ\else\textdegree\fi{}K for impurity concentrations ${N}_{I}\ensuremath{\le}{10}^{15}$ ${\mathrm{cm}}^{\ensuremath{-}3}$, but overestimates the mobility for higher ${N}_{I}$ or lower $T$. The eventual failure of the analysis may be attributed to the neglect of carrier-carrier and interband ionized impurity scattering.

Experimental Investigation of Conduction Band of GaSb

The conduction band of GaSb was investigated by making the following measurements on a number of $n$-type samples with different carrier concentrations: (1) Hall effect and conductivity between 1.5\ifmmode^\circ\else\textdegree\fi{}K and 370\ifmmode^\circ\else\textdegree\fi{}K; (2) the change of resistance and Hall effect of the samples under hydrostatic pressure (up to 14\ifmmode\times\else\texttimes\fi{}${10}^{9}$ dynes/${\mathrm{cm}}^{2}$) at room temperature; and (3) the change of resistance due to uniaxial stress between 77\ifmmode^\circ\else\textdegree\fi{}K and 370\ifmmode^\circ\else\textdegree\fi{}K. Our data can be explained on the basis of a double conduction band for this material with the lowest band-minimum in the center of the Brillouin Zone characterized by spherical constant-energy surfaces and the next minima along [111] directions in $k$-space characterized by ellipsoidal constant-energy surfaces. Our data can be further interpreted by assuming that the deformation potentials for the two bands in GaSb are similar to those of the corresponding bands in $n$-type germanium. The energy separation of the two bands at room temperature is estimated to be about 0.08 ev. The observed temperature dependence of the piezoresistance could be explained by assuming that the energy separation $\ensuremath{\Delta}E$ changes with temperature at a rate $(\frac{d\ensuremath{\Delta}E}{\mathrm{dT}})\ensuremath{\approx}\ensuremath{-}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ ev/\ifmmode^\circ\else\textdegree\fi{}C. Throughout the analysis, the relaxation times for the electrons in the two bands were assumed to be independent of energy.

Mobility of Holes and Electrons in High Electric Fields

The field dependence of mobility has been determined for electrons and holes in both germanium and silicon. The observed critical field at 298\ifmmode^\circ\else\textdegree\fi{}K beyond which $\ensuremath{\mu}$ varies as ${E}^{\ensuremath{-}\frac{1}{2}}$ is 900 volts/cm for $n$-type germanium, 1400 volts/cm for $p$-type germanium, 2500 volts/cm for $n$-type silicon, and 7500 volts/cm for $p$-type silicon. These values of critical field are between two to four times those calculated on the basis of spherical constant energy surfaces in the Brillouin zone. A saturation drift velocity of ${6(10)}^{6}$ cm/sec is observed in germanium which is in good agreement with predictions based on scattering by the optical modes. Data on $n$-type germanium at 20\ifmmode^\circ\else\textdegree\fi{}K show a range over which impurity scattering decreases and the mobility increases with field until lattice scattering dominates as at the higher temperatures.