We discuss in detail the effects of Landau quantization on the quantum dynamics and statistical thermodynamics of Dirac-like materials in a high magnetic field. This discussion addresses the Group VI Dichalcogenides and Graphene in particular, but it also extends to the "Diced Lattice" as well as Silicene and Topological Insulators, etc. We also treat the role of nanostructures in Dirac-like materials, including quantum dots, quantum wires and a lattice of anti-dots.
Starting with the derivation of the Landau-quantized "relativistic" Green's function for Dirac-like materials in position and momentum representations, as well as in eigenfunction representation, the associated energy spectrum in a high magnetic field is discussed. Furthermore, we develop the Green's functions for a model quantum dot in Graphene and a model quantum wire in the Group VI Dichalcogenides, examining their Landau quantized spectra as well. We also treat the case of a periodic lattice of anti-dots in the Dichalcogenides,
tracing in detail the development of Landau minibands, with an explicit derivation of the Green's function for "relativistic" carriers in the lattice in a high magnetic field.
In addition to the above-cited quantum dynamical stndies of "Dirac-like" materials (and spectra) in a high magnetic field we have also examined their statistical thermodynamics, developing the associated thermodynamic Green's function and spectral weight matrix subject to Landau quantization. With this we have determined the Helmholtz Free Energy, the Grand Partition Function as well as the ordinary Partition Function and the Entropy for the Dichalcogenides and the Diced Lattice in a high magnetic field explicitly; and have employed these to evaluate the magnetization of the "Dirac-like" materials in both degenerate and non degenerate statistical regimes, replete with de Haas-van Alphen oscillatory phenomenology (in the degenerate case).
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