Landau States Research Articles

Correlation Energy of an Electron Gas in the Quantum Strong-Field Limit

The ground state of an electron gas in an intense magnetic field is studied using a wave function of the product form ${\ensuremath{\Psi}}_{0}^{B}\ensuremath{\Phi}$. The correlation factor ${\ensuremath{\Psi}}_{0}^{B}$ is taken to be the ground-state solution of a charged Bose gas and $\ensuremath{\Phi}$ a determinant of single-particle Landau states. In the quantum strong-field limit so that only the lowest Landau state is populated, the correlation energy is computed using the cluster-expansion technique and a variational determination of the boson energy. Numerical results obtained are lower than those derived under the random-phase approximation.

Spin-Orbit Effects in Crossed Electric and Magnetic Fields;Γ7Band of Wurtzite-Type Crystals

The ${\ensuremath{\Gamma}}_{7}$ conduction band of the wurtzite-type II-VI compounds has a spin-orbit term linear in k. Effects of this spin-orbit term in crossed electric and magnetic fields are investigated on the basis of a one-band formalism. The effective Hamiltonian is solved for four different cases, according to the strength of the crossed electric and magnetic fields: (A) Strong magnetic field and weak electric field---this is essentially the simple band case; (B) weak magnetic field and weak electric field (the coupling between cyclotron motion and spin states via the spin-orbit term plays a dominant role in determining the energy spectrum); (C) strong magnetic field and strong electric field. (The major contribution to the spin-orbit interaction comes from a term representing the influence of transverse drift motion upon spin states. As a result, the spin splitting exhibits a nearly linear dependence on the electric field, and the direction of the spin axis tends toward that of the electric field with increasing electric field); (D) weak magnetic field and strong electric field [both of the spin-orbit effects, which are predominant either in the case (B) or in the case (C), are equally important, and perturbation theory is not applicable. Variational solutions have amplitudes distributed over many Landau and spin states, so that the selection rule for the intraband transitions is relaxed.] The conditions for observing these spin-orbit effects by means of intraband transitions are discussed for the actual II-VI compounds. It is found that spin-orbit effects may possibly be observed in magnetic dipole transitions under the condition of case C for CdS and ZnS, as well as in electric and magnetic dipole transitions under the condition of case A or C for CdSe. In the strong electric field of case C, the transverse drift velocity is one or two orders of magnitude larger than the velocity of sound in the crystal. Hence, phonon clouds build up around an electron to cause broadening of the resonance line. This can be avoided by carrying out the resonance experiment before the phonon clouds build up, i.e., by employing a pulsed transverse electric field.

The Theory of Low‐Field Oscillations of the Surface Impedance in Metals

AbstractMagnetic surface states on cylindrical parts of the Fermi surface and their relation to volume Landau states are discussed. The scattering of the surface states by a rough surface is calculated quantum mechanically, giving the classical scattering rate in a certain limiting case. Transitions between surface states give rise to small oscillations of the surface impedance, which are calculated for arbitrary orientations of the cylinder and the magnetic field, starting with the electric field distribution in the case of anomalous skin effect. The results are compared with experimental data in indium. More elaborate numerical calculations of the transition matrix elements and the surface scattering rate are required to determine the parameters of surface scattering unambiguously.

The effects of the electron - lattice interaction on the free-carrier magneto-optics of semiconductors

The effects of the electron - lattice interaction on the free-carrier magneto-optical properties of semiconductors are treated by developing a model due to Johnson and Larsen based on one-phonon excitation of magnetic Landau states. The main result is to explain a large discontinuous broadening observed by Summers, Harper and Smith in InSb at magnetic fields H given by eH/mc = ωo, the lattice long-wave longitudinal optic frequency.

Effective-Mass Approximation for Electrons in Crossed Electric and Magnetic Fields

A criterion of applicability of the effective-mass approximation to the dynamics of a Bloch electron in crossed electric and magnetic fields is derived based on a treatment of Luttinger and Kohn. The requirements necessary for the diagonalizability of the Hamiltonian for an electron in a periodic crystal field in the presence of a perturbing magnetic field and an electric potential are examined. These requirements lead to a restriction on the ratio of the electric and magnetic fields. The validity of the effective-mass approximation is also considered for higher Landau states in a magnetic field as well as for lowest states of an impurity ion.

On a Tauberian theorem of Landau

T(x) = A(-) = xlog x+ bx+ o(x) nfix n with b a constant, theni (1) c1x < A(x) < c2x, where c1 and c2 are positive constants. This gives Tschebyschef's theorem if Mwe take A(x) = A(x) and use the fact that log [x]! = x log x x + O(log x). Landau states that if only the condition T(x) = x log x + O(x) is assumed, then (1) does not follow, a remark whose incorrectness was proved by H. N. Shapiro [4]. Landau then goes on to prove [3, pp. 598-604] that if Received by the editors June 14, 1957 and, in revised form, November 25, 1957 and February 24, 1958. 1 Here and in the sequel, (x), A(n), ,(n), and @(x) denote the usual functions of prime number theory. 693