Self-Similar Optical Absorption Spectra in High Magnetic Fields

Abstract

Employing a non-perturbative multi-band approach within empirical tight-binding theory, we show that the magneto-absorption spectrum of a periodic two and three-dimensional electron system possesses a structure that overall reflects the fractal Hofstadter butterfly eigenvalue spectrum. For small magnetic fields, the standard magneto-optical selection results of envelope theory are recovered. We predict that the Hofstadter butterfly can be directly observed by interband magnetoabsorption from an undoped GaAs quantum well beneath a periodic array of stressors.1. Preliminary Remarks This paper is dedicated to my dear colleague Roland Zimmermann whom I greatly admire and value. Over the many years where I have known him, I have greatly profited directly and indirectly from his brilliant and critical mind, his expertise in many areas of solid state physics, but as well from his personal integrity and warmth.This paper is the result of an ongoing research in a project where Roland Zimmermann served as reviewer. It demonstrates that the nonperturbative treatment of interaction effects uncovers interesting and unusual physical effects. While the details of this paper lie outside of Roland Zimmermann's main interests, it is this central message that overlaps with much of his outstanding work in the past and -so we all hope -for many years to come. IntroductionSince the discovery of the Hofstadter butterfly [1] the behaviour of electrons that are subject to a magnetic field and a periodic potential of comparable energy or length scale has challenged both experimentalists and theorists.Theoretically, the usual perturbative treatment of the magnetic field or the so-called Peierls substitution "ðkÞ ! "ðÀir þ eA=ð hcÞÞ in an envelope function approach [2] breaks down when the dimensions of the zero field periodicity cell and the cyclotron radius become comparable to each other. The main challenges in a correct description of this situation lie in the fact that (i) the magnetic field destroys the original periodicity of the problem and (ii) the vector potential associated with a constant magnetic field A k x is an unbound function that hampers the use of any kind of periodic boundary conditions. Several approaches have been developed to deal with this problem nonperturbatively [3][4][5][6]. So far, most theoretical studies focused on the electronic energy spectrum [4][5][6] and the Hall conductance [4] within one-band or many-band situations phys. stat. sol. (b) 234, which both reflect the fractal eigenvalue spectrum of the Hofstadter butterfly. It has not been clarified so far, however, whether the optical absorption spectrum is generally fractal as well or whether the robustness of Kohn's theorem [7] together with optical selection rules will mask the self-similar eigenvalue spectrum. In this paper, we present a fully non-perturbative study of the optical absorption of periodic systems in constant magnetic fields. Particularly, we will show that the optical spectrum of two-dimensional periodic systems in magnetic fields exhi...

Optical properties of the Hofstadter butterfly in the moiré superlattice

“…For TBG, the optical spectrum at zero magnetic field was recently studied theoretically [27][28][29][30][31], and experimentally [27,32], while the magneto-optical property remains to be explored. In the literature, the optical absorption of the Hofstadter butterfly was studied for a simple square lattice [33], while a detailed study is needed to specify the optical selection rule in the recursive spectrum.…”