Probing two-subband systems in a quantizing magnetic field with non-equilibrium phonons

We propose to use phonon absorption spectroscopy to study many-body gaps and phases of two-subband heterostructures in the quantum Hall regime. Implications of the spinorbit interaction for phonon absorption in this system are considered. Phonon spectroscopy is a powerful experimental technique that has been successfully used in the study of the properties of a two-dimensional (2D) electron gas in the quantum Hall regime [1]. It allows probing of the states with non-zero momentum, which are not accessible by either cyclotron resonance or photoluminescence methods. We propose to use acoustic-phonon spectroscopy to study many-body effects in a two-subband quasi-2D system, formed at a single semiconductor heterojunction and subjected to a quantizing magnetic field. We show that the formation of many-body gaps and the existence of different spin phases in this system result in a striking difference in the magnetic field dependence of the phonon absorption rate at different filling factors. Acoustic-phonon scattering of 2D electrons in a quantizing magnetic field is strongly suppressed due to the dispersionless structure of the Landau levels (LLs). In a two-subband system, however, a sharp enhancement of the electron-phonon interaction occurs when the two LLs corresponding to different size-quantization subbands (we consider the second LL of the first subband and the first LL of the second subband only) approach one another, and the separation between these energy levels, ∆ = ∆12 − h̄ωc, becomes close to h̄s/l. Here h̄ωc is the cyclotron energy, ∆12 is the intersubband splitting, s is the velocity of sound and l is the magnetic length. This electron-phonon interaction enhancement should manifest itself in the magnetic field dependence of the phonon absorption rate and dissipative conductivity. At filling factor slightly greater than two, when the electron-electron interaction between the particles belonging to either of the intersecting LLs can be neglected, a double-peak structure in the magnetic field dependence of the phonon-mediated transport properties of the two-subband system has been predicted [2]. With increasing electron density, the repulsion between the levels due to electron-electron exchange interactions opens a gap in the interlevel excitation spectra. At filling factor ν = 3 this gap exists at all values of the bare level splitting ∆ and can well exceed h̄s/l. As a result, the double-peak structure transforms into a singlepeak one (see Figure 1). By far the most interesting case is when the electron filling factor ν = 4. When the cyclotron energy is close to the intersubband splitting the system can be mapped onto a four-level electron system with an effective filling factor ν∗ = 2. The ground state is either a ferromagnetic state or a spin-singlet state, depending on the values of the interlevel splitting and Zeeman energy ∆Z . The electron-electron interaction renormalizes the phase boundaries, which results in the existence of a ferromagnetic phase even for ∆Z = 0. The excitation spectrum has a gap for any value of the interlevel splitting ∆ and Zeeman Probing two-subband systems in a quantizing magnetic field with non-equilibrium phonons 2 splitting ∆Z. This results in strong suppression of the electron-phonon interaction. The rate of phonon absorption by the considered quasi-2D electron system at ν = 4 again has a doublepeak structure as a function of level splitting and a steplike structure as a function of Zeeman splitting (see Figure 2). In the absence of the spin-orbit interaction, the transitions between levels with different spin indices are forbidden because the electron-phonon interaction conserves the electron spin. The spin-orbit interaction allows such transitions. Matrix elements of the spin-orbit interaction Hamiltonian HSO between levels differing by subband number, LL number, and spin projection on the normal to the structure plane, are given by following expressions [3]: 〈η,n,kx,−1/2 |HSO|μ,n−1,kx,+1/2〉 = iC 1 lH √ 2n, 〈η,n,kx,+1/2 |HSO|μ,n−1,kx,−1/2〉 = A 1 lH √ 2n, 〈η,n−1,kx,−1/2 |HSO|μ,n,kx,+1/2〉 = A 1 lH √ 2n, 〈η,n−1,kx,+1/2 |HSO|μ,n,kx,−1/2〉 = − iC 1 lH √ 2n, (1)