Landau level spectroscopy in current solid state physics

Abstract

Landau level spectroscopy plays an important role in modern condensed-matter physics. In this technique, electrons in a solid are subjected to quantizing magnetic fields and probed experimentally, often through optical methods. Direct and detailed insights into the electronic properties of crystalline materials are obtained, particularly the properties related to their band structure. Landau level spectroscopy enables the precise extraction of key parameters such as effective mass, carrier density, mobility, and band gap, and serves as a powerful tool for studying interactions between electrons and other quasiparticles in solids. Over its more than seventy-year history, Landau level spectroscopy has been applied mainly to semiconductors and semimetals. Today, its scope also includes graphene-based systems, surface and bulk states in topological materials, and other emergent systems with a narrow or vanishing band gap. In this work, we review the fundamentals of Landau level spectroscopy and illustrate them with selected examples from the literature.

Figures (14)

• Figure 1: One of the first reported cyclotron resonance measurements, performed on germanium Dresselhaus:1955b. In this experiment, absorption of a monochromatic microwave (24 GHz) in germanium kept at low temperatures (4 K) was traced as a function of the applied magnetic field. While we would expect to see only electrons or only holes, both kinds of carriers were seen in the experiments. The authors interpreted this as a result of their sample being ionized by microwaves. Reprinted with permission from Dresselhaus:1955b, copyright (1955) by the American Physical Society.
• Figure 2: Modulation of mid-infrared transmission through a crystal of InSb, induced by the magnetic field, was one of the very first optical experiments showing Landau quantization in a solid via interband optical excitations. The plot shows relative magneto-transmission for selected values of magnetic field. The dip corresponds to an interband inter-Landau level transition, which is a remarkable observation at such low magnetic fields ($2-4$ T) and at room temperature. Reprinted with permission from ZwerdlingPR57, copyright (1957) by the American Physical Society.
• Figure 3: Cut-away drawing of an early experimental setup developed for Landau level spectroscopy of semiconductors in late 1950's, prior to Fourier transform spectroscopy was introduced during 1960s. It is composed of a radiation source, a grating monochromator, a cryostat and an electromagnet. This setup was used to study interband excitations in Landau-quantized germanium ZwerdlingPR59. Reprinted with permission from ZwerdlingPR59, copyright (1959) by the American Physical Society.
• Figure 4: Cyclotron motion of an electron in an arbitrary band. The electron encircles an area $A$, whose energy dependence defines the effective cyclotron mass, see Eq. \ref{['classicalCRmass']}.
• Figure 5: Magneto-absorbance (the real part of optical conductivity) of highly doped graphene measured at low temperatures. The observed transition is linear in $B$, and it is a quasi-classical cyclotron resonance. The intensity and width of the mode is slightly modulated with $B$, which is an indication that the regime of Landau quantization is approached. In this regime, the $\sqrt{B}$ scaled transitions are shown in gray dashed lines, and particular filling factors indicated by red dashed lines; see OrlitaNJP12 for details.