Microscopic theory of the cyclotron-resonance line shape of an interacting, disordered two-dimensional electron gas beyond the leading-order, integer-filling, strong-field limit, and the origin of anomalous cyclotron-resonance splittings at intermediate magnetic fields

In a two-dimensional electron gas (2DEG) subjected to a strong perpendicular magnetic field, far-infrared cyclotron-resonance (CR) absorption probes the interplay of electron-electron interactions, disorder, and Landau quantization. Kallin and Halperin [Phys. Rev. B 31, 3635 (1985)] gave a controlled theory of the CR line at integer filling factors in the strong-field limit, treating correlations exactly to leading order in the Coulomb energy divided by the cyclotron energy and impurity scattering self-consistently, and showed that this theory does not explain the narrow split CR lines observed at intermediate fields where the two energy scales are comparable. The unresolved problem is to construct a quantitative microscopic theory of the CR absorption of a disordered, interacting 2DEG valid beyond that restricted regime - including higher-order correlation corrections, partially filled Landau levels, finite temperature, and nonperturbative disorder effects - and to determine from microscopic ingredients which mechanisms (disorder-mediated coupling to finite-wave-vector magnetoplasmon modes with roton-like minima, spin-dependent interactions including spin-orbit-induced mixing of density and spin-density excitations, electron-phonon coupling, valley degeneracy) produce the anomalous CR splittings observed in GaAs/AlGaAs, Si inversion layers, and AlGaN/GaN, reproducing the observed line shapes, linewidths, and their dependence on filling factor, density, and disorder.

Background

A two-dimensional electron gas (2DEG) - electrons confined to a plane at the interface of a semiconductor heterostructure such as GaAs/AlGaAs, in a Si metal-oxide-semiconductor field-effect-transistor (MOSFET) inversion layer, or in an AlGaN/GaN heterojunction - placed in a perpendicular magnetic field $B$ has its kinetic energy quantized into highly degenerate Landau levels separated by the cyclotron energy $\hbar\omega_c$ , where $\omega_c=eB/m^*c$ is the cyclotron frequency ( $e$ the electron charge, $m^*$ the effective mass, $c$ the speed of light). Far-infrared light at $\omega\approx\omega_c$ excites cyclotron resonance (CR), the standard probe of $m^*$ . Two energy scales compete: the Coulomb energy $E_{Co}=e^{2}/\epsilon l_{0}$ (with $\epsilon$ the dielectric constant and $l_{0}=(\hbar c/eB)^{1/2}$ the magnetic length, the characteristic radius of a cyclotron orbit) and $\hbar\omega_c$ . Their ratio $E_{Co}/\hbar\omega_c$ falls as $1/\sqrt{B}$ , so 'strong field' means $E_{Co}\ll\hbar\omega_c$ . Kohn's theorem [W. Kohn, Phys. Rev. 123, 1242 (1961)] states that for a translationally invariant system of carriers with a parabolic band and arbitrarily strong (short-range) interactions, the $q=0$ CR frequency is unrenormalized by interactions; in real 2DEGs, broken translational invariance (disorder, finite well width) and nonparabolic band structure open the door to observable many-body shifts and linewidth effects.

At integer filling factor $\nu$ (the number of filled Landau levels, $\nu=2\pi l_{0}^{2}n_{e}$ for electron density $n_{e}$ ), the neutral inter-Landau-level excitations are well described as magnetoexcitons, or magnetoplasmons: electron-hole pairs bound by Coulomb attraction, labeled by an in-plane wave vector $\mathbf{k}$ , with a dispersion $E(\mathbf{k})$ that deviates from $\hbar\omega_c$ by terms of order $E_{Co}$ [C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655 (1984)]. The dispersion develops a 'roton-like' minimum at $|\mathbf{k}|l_{0}\sim1$ ; light-scattering experiments directly determined the resulting roton densities of states [A. Pinczuk et al., Surf. Sci. 229, 384 (1990)]. Disorder (residual ionized impurities, interface roughness) broadens the CR line. Standard treatments treat impurity scattering in the self-consistent Born approximation (SCBA, a diagrammatic resummation in which the impurity-averaged self-energy is determined self-consistently) while neglecting interactions, or treat interactions in approximations such as the random-phase approximation that are uncontrolled at large $\mathbf{k}$ in strong fields.

Kallin and Halperin [Phys. Rev. B 31, 3635 (1985)] combined the two: their self-consistent exciton approximation (SCEA) treats electron-electron interactions exactly to leading order in $(e^{2}/\epsilon l_{0})/\hbar\omega_c$ by working in the magnetoexciton basis with the pure-system dispersion $E(\mathbf{k})$ , and treats impurity scattering of excitons self-consistently, including both self-energy and vertex corrections (corrections to the optical matrix element that cancel unphysical divergences of low-order impurity perturbation theory). At integer filling and $T\approx0$ this yields CR lines shifted to lower frequency with correlation-induced narrowing, linewidths of order $1\ \mathrm{cm}^{-1}$ for realistic GaAs heterojunction parameters (in agreement with experiment), a strong dependence of the linewidth on the spatial range of the impurity potential, and the prediction that finite- $\mathbf{k}$ magnetoplasmon branches are strongly damped while their energies remain near the pure-system dispersion. A Halperin-Lax-type rare-fluctuation estimate (Halperin and Lax's theory of disorder-induced localized tail states) suggests an additional, exponentially small linewidth contribution $\propto\exp(-\gamma/U_{2})$ from impurity-localized excitons, negligible for realistic impurity strength $U_{2}$ .

The SCEA regime is, however, narrow. Most experiments have $E_{Co}/\hbar\omega_c\sim1$ (intermediate fields), partially filled Landau levels, or nonzero temperature. Most strikingly, narrow split CR lines are observed at intermediate fields, in GaAs/AlGaAs heterojunctions [G.L.J.A. Rikken et al., Surf. Sci. 170, 160 (1986)], in Si inversion layers at low densities [J.-P. Cheng et al., Surf. Sci. 196, 268 (1988)], and - with splittings up to 2 meV - in AlGaN/GaN heterojunctions [S. Syed et al., Phys. Rev. B 67, 241304(R) (2003)], while the SCEA analysis showed that coupling of the $q=0$ CR mode to broadened finite- $q$ magnetoplasmon modes, within its domain of validity, does not reproduce the observed narrow splittings, and that higher-order Hartree-Fock-type corrections to the exciton dispersion are too small to bring a finite- $q$ mode into resonance with $\omega_c$ for $E_{Co}/\hbar\omega_c\le 2$ . Kallin and Halperin therefore identified spin-dependent interactions, electron-phonon coupling, valley degeneracy (the twofold orbital degeneracy of the Si conduction band), other many-body effects, or different impurity-potential models as candidate mechanisms to be determined, together with the open extensions of the theory to higher order in $E_{Co}/\hbar\omega_c$ , to noninteger filling factors, and to finite temperature.

Problem Statement

Construct a quantitative, controlled microscopic theory of the far-infrared cyclotron-resonance absorption of a disordered, interacting two-dimensional electron gas in a perpendicular magnetic field, valid outside the strong-field, integer-filling, zero-temperature regime in which the self-consistent exciton approximation of Kallin and Halperin (1985) is controlled, and use it to determine the origin of the anomalous cyclotron-resonance splittings and linewidth anomalies observed at intermediate fields where $E_{Co}/\hbar\omega_c\approx1$ . Because $(e^{2}/\epsilon l_{0})/\hbar\omega_c$ is of order unity in this intermediate-field regime, 'controlled' here does not presuppose that ratio as a small expansion parameter: it requires the approximation scheme to be specified explicitly with an assessment of its accuracy - a stated expansion parameter and error estimate, or, where no such parameter exists, a stated regime of validity and a quantitative error estimate - so that the resulting predictions can be judged.

Concretely, the problem - the five linked limitations identified in the 1985 paper, unified by this common objective - asks for: (1) the quantitative form of the many-body corrections to the magnetoexciton dispersion, to the coupling of the $q=0$ cyclotron mode to finite-wave-vector magnetoplasmon modes, and to the CR line shapes, beyond leading order in $(e^{2}/\epsilon l_{0})/\hbar\omega_c$ ; (2) identification, from microscopic ingredients, of the mechanism that produces the narrow split CR lines observed at intermediate fields in GaAs/AlGaAs heterojunctions, Si inversion layers, and AlGaN/GaN heterojunctions - including whether it is disorder-mediated coupling to the roton-like finite- $q$ minimum of the magnetoplasmon dispersion, spin-dependent interactions such as spin-orbit-induced mixing of density and spin-density excitations, electron-phonon coupling, valley degeneracy, or a combination - with a theory that reproduces the observed line shapes, splittings, and linewidths and their dependence on filling factor, electron density, and disorder; (3) the quantitative role of nonperturbative disorder effects (rare impurity fluctuations, localization of excitons) in the CR linewidth, and how they compete with magnetic-field inhomogeneity, temperature, and higher-order electron-electron terms; (4) comparison with systematic experiments at integer and noninteger filling factors in which electron density and impurity disorder are varied independently, sufficient to test the predicted line shifts, impurity-range dependence, and wave-vector-dependent linewidth growth, and to infer the impurity-potential spectrum of the devices; and (5) the extension to partially filled Landau levels and finite temperature, where screening, additional damping channels, and collective modes may qualitatively change the absorption. The mechanism identification sought in (2) is established by a quantitative reproduction of the measured line shapes, splittings, and linewidths and their dependence on filling factor, electron density, and disorder, not by qualitative agreement alone; and (4) compares the predicted line shifts, impurity-range dependence, and wave-vector-dependent linewidth growth with the systematic experiments available, as far as those data allow.

References

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