Coupling of Spin Waves with Charge- and Spin Density Excitations in Spin-Polarized Quantum Wells

The coupling of spin waves with charge- and spin-density waves is shown to be induced by a spin-dependent interaction in a quantum well, which is spin polarized by a dc magnetic field at an angle u to the symmetry axis. The mixing of the plasmonic and magnonic modes, which occurs for both intra- and intersubband transitions, depends on the coupling constant of the spin-spin interaction, the tilt angle u, and the initial spin polarization z. @S0163-1829~99!08247-8# An intense experimental effort employing infrared absorption and inelastic light-scattering techniques has been focused on probing the excitation spectrum of an electron gas confined to a quasi-two-dimensional semiconductor heterostructure ~2DEG!. 1,2 The observed absorption peaks indicate that linearly independent charge-density waves ~CDW’s!, spin-density waves ~SDW’s!, and spin waves ~SW’s ! propagate at frequencies corresponding to the poles of the appropriate response functions. The first two excitations are generated by fluctuations in the local density of particles with a given spin, while the spin waves are associated with spin-flip processes. When a dc magnetic field is applied perpendicular to the 2D layer, the energy levels within each subband are determined by the ratio of the Zeeman splitting 2g*B and the cyclotron energy \v c5\eB/m*c (g* is the effective g value and m* is the effective mass! .I f 2g*B!\v c ~as in GaAs heterostructures!, each subband is characterized by a sequence of Landau levels, each of which displays a small spin splitting. 3 In the opposite limit, 2 g*B@\v c ~as can occur for dilute magnetic semiconductor structures!, the main splitting within each subband is into widely separated spin-up and spin-down components. The cyclotron energy gives rise to small splittings of the spin subbands into Landau-level ladders. The SW’s are not coupled to the CDW and the SDW when the magnetic field is normal to the 2D layer ~considered here as the x-z plane!. The shift of the collective excitation frequency from the single-particle transition is a consequence of the many-body interaction. Theoretical investigations of the elementary spectrum of a 2D electron gas in a normal magnetic field have shown that the existence of the CDW and intersubband SDW is a consequence of the mean electrostatic field, the random-phase approximation ~RPA! of the Coulomb repulsion. 4 Intrasubband SDW and both intra- and intersubband SW’s appear only when a spin-dependent interaction is included. 5 The former is negligible in GaAs where g* is very small but can give observable effects in dilute magnetic semiconductors, where the effective gyromagnetic factor is enhanced by the electron interaction with the magnetic ions to values up to a hundred times its band value. In this paper, we consider the applied dc magnetic fieldB W at an angle u to the symmetry axis of the dilute magnetic semiconductor well and show that the tilt of the magnetic field produces the coupling of the spin- and charge-density waves with the spin waves. The excitation frequencies for the collective modes are derived within the RPA approximation by using the equation of motion for the one-electron distribution function. Analytic results are obtained for small values of the inclination angle u. The dependence of the collective-mode frequencies on the equilibrium polarization and u are studied in the long-wavelength limit. The electron state in a quantum well of width L is a plane wave of wave vector k ~in the x ˆ-z plane! modulated by an envelope function that reflects the y ˆ-axis confinement. The associated field operator is