PHYSICAL REVIEW 8 VOLUME 43, NUMBER 5 15 FEBRUARY ]991-I SeÃ-consistent Landau-level broadening by acoustic phonons in two-dimensional electron systems B. Tanatar and M. Singh Department of Physics, The Uniuersity of Western Ontario, London, Ontario, Canada X6A 3ICT A. H. Macoonald Department of Physics, Indiana Uniuersity, Bloomington, Indiana 47405 (Received 8 June 1990) We have studied the contribution of electron-phonon scattering to the Landau-level broadening of a two-dimensional electron gas in a strong perpendicular magnetic field. The results depend qual- itatively on the contribution to the Landau-level width from electron-disorder scattering and it is necessary to treat both contributions self-consistently. In high-mobility samples, electron-phonon scattering produces a peak in the density of states at the Fermi energy. I. INTRODUCTION The quantization of the kinetic energy of a two- dimensional electron gas' (2D ECr) in a perpendicular magnetic field into Landau levels is responsible for many anomalies. Usually the strength of an anomaly depends on the degree to which the Landau levels are broadened by disorder and anomalies disappear when the Landau- level width I becomes larger than the Landau-level sepa- ration Ace, . For this reason, an understanding of the properties of a 2D EG in a strong perpendicular magnet- ic field often begins with an understanding of the broadening of its Landau levels. At low temperatures, Landau-level broadening in high-quality molecular- beam-epitaxy (MBE) -grown samples is usually dominat- ed by scattering from remote ionized impurities. At high temperatures (T+ 50 K), the electron scattering is dominated by interactions with LO phonons. For inter- mediate temperatures (4 «T «40 K), electron scattering in high-mobility samples can be dominated by interac- tions with acoustic phonons. ' ' Previous work on Landau-level broadening by acoustic-phonon scattering relied on an elastic approximation in which the frequen- cies of the relevant phonons were implicitly assumed to be small compared with both I and k&T. In this paper we report on a study of the contribution of interactions with acoustic phonons to Landau-level broadening in which this approximation is not made. We restrict our attention to the strong-field limit where the Fermi level lies within the lowest Landau level and we can ignore Landau-level mixing by disorder or by virtual-phonon emission or absorption. In this limit, the elastic approxi- mation can never be strictly justified. We find that for high-mobility samples, the contribution of electron — acoustic-phonon interactions to Landau-level widths can be comparable to the disorder contribution even at low temperatures. In the low-temperature limit, a strong analogy exists between the acoustic-phonon contribution to the electron self-energy io the present problem and the phonon contri- bution to the self-energy of electrons in metals. ' The role played by the Fermi energy or the bandwidth in the metallic case is played here by the Landau-level width I". We find that there is an analog for the strong-field 2D EG of the electron-phonon enhancement of the Fermi-level density of states in metals. For k~T (&I, the contribu- tion to the electron self-energy from the emission and ab- sorption of virtual phonons produces a peak in the density-of-states at the Fermi level. The peak becomes more pronounced in high-mobility samples with smaller Landau-level widths. The peak at the Fermi level can strongly distort the shape of the density of states, espe- cially when the Fermi level moves away from the middle of the Landau level. The perturbative treatment of the phonon contribution to the electron self-energy is based on an expansion in which the sma/l parameter, at least at low temperatures, is an energy characteristic of the phonon system (see below) divided by I . In the absence of a contribution to I from disorder the perturbative approach breaks down at low temperatures. In order to treat the low- temperature regime correctly it is essential to include a contribution to the electron self-energy from disorder and to treat the two contributions self-consistently. This paper is organized as follows. In Sec. II we derive the expressions for the self-energy on which our calcula- tion is based and discuss the methods used to obtain self- consistent solutions of the equations which arise. In Sec. III we present our results. We define a quality which is analogous to the a F(co) function often' used to discuss the phonon contribution to electron self-energies in met- als. With this definition the strong analogy with the me- tallic case becomes obvious. We discuss the importance of the magnetic-field strength and of the width of the 2D EG in determining the maximum relevant acoustic- phonon frequency, and the importance of electron- phonon coupling. Some concluding remarks are present- ed in Sec. IV. II. THEORY A. Model Qur treatment of the role of electron-phonon interac- tions in a quasi-two-dimensional electron system with a 4308 199I The American Physical Society