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We consider the polaron dynamics driven by Froḧlich type, long wavelength dominated electron-phonon interaction at zero temperature, for three different semi-metals: single and bilayer graphene, and semi-Dirac, all grown on polar substrates such as, SiC . Single layer graphene (henceforth called SL graphene), bilayer graphene (henceforth called BL graphene), and semi-Dirac have two dimensional band-structures with point Fermi surfaces in their natural undoped conditions. When these materials are grown on polar substrates, their electrons can interact with the optical phonons (LO) at the surface of the substrates. That gives rise to the possibility of polaron formation in the context of these semi-metals, although they themselves are non-polar. Starting from the Froḧlich type electron-phonon interaction Hamiltonian, perturbation theory is employed to calculate the self energy of the electron due to polaron formation for the three aforementioned systems. The electron self energy, or the polaron energy, calculated analytically for BL graphene, is shown to vary linearly with the electron momentum for small electron momenta. Whereas for ordinary polar crystals (both two and three dimensional), for small electron momentum, the polaron energy is quadratic leading to the mass correction of the electron, for BL graphene the polaron energy disperses linearly, rendering the massive BL graphene electrons effectively massless. Energies for Froḧlich polarons in SL graphene and semi-Dirac on polar substrates, are numerically evaluated. Also, the electron relaxation rate, related to the imaginary part of the analytically continued electron self energy expression, is calculated for the three systems.

We consider the polaron dynamics driven by Froḧlich type, long wavelength dominated electron-phonon interaction at zero temperature, for three different semi-metals: single and bilayer graphene, and semi-Dirac, all grown on polar substrates such as, SiC. Single layer graphene (henceforth called SL graphene), bilayer graphene (henceforth called BL graphene), and semi-Dirac have two dimensional band-structures with point Fermi surfaces in their natural undoped conditions. When these materials are grown on polar substrates, their electrons can interact with the optical phonons (LO) at the surface of the substrates. That gives rise to the possibility of polaron formation in the context of these semi-metals, although they themselves are non-polar. Starting from the Froḧlich type electron-phonon interaction Hamiltonian, perturbation theory is employed to calculate the self energy of the electron due to polaron formation for the three aforementioned systems. The electron self energy, or the polaron energy, calculated analytically for BL graphene, is shown to vary linearly with the electron momentum for small electron momenta. Whereas for ordinary polar crystals (both two and three dimensional), for small electron momentum, the polaron energy is quadratic leading to the mass correction of the electron, for BL graphene the polaron energy disperses linearly, rendering the massive BL graphene electrons effectively massless. Energies for Froḧlich polarons in SL graphene and semi-Dirac on polar substrates, are numerically evaluated. Also, the electron relaxation rate, related to the imaginary part of the analytically continued electron self energy expression, is calculated for the three systems.

Digital PCR, a state-of-the-art nucleic acid quantification technique, works by spreading the target material across a large number of partitions. The average number of molecules per partition is estimated using Poisson statistics, and then converted into concentration by dividing by partition volume. In this standard approach, identical partition sizing is assumed. Violations of this assumption result in underestimation of target quantity, when using Poisson modeling, especially at higher concentrations. The Poisson-Plus Model accommodates for this underestimation, if statistics of the volume variation are well characterized. The volume variation was measured on the chip array based QuantStudio 3D Digital PCR System using the ROX fluorescence level as a proxy for effective load volume per through-hole. Monte Carlo simulations demonstrate the efficacy of the proposed correction. Empirical measurement of model parameters characterizing the effective load volume on QuantStudio 3D Digital PCR chips is presented. The model was used to analyze digital PCR experiments and showed improved accuracy in quantification. At the higher concentrations, the modeling must take effective fill volume variation into account to produce accurate estimates. The extent of the difference from the standard to the new modeling is positively correlated to the extent of fill volume variation in the effective load of your reactions.

In graphene the energy momentum dispersion has a point Fermi surface and is linear at the zone boundary. To actually discover a feature in a band structure that provides the quasiparticle dispersion of a new and unexpected type is rare, the discovery of a \"semi Dirac'' dispersion pinned to the Fermi energy is a very recent example. Pardo et al. reported such a finding in ultrathin (001) VO2 layers embedded in TiO2. This new point Fermi surface system, dubbed 'semi-Dirac,' is a hybrid of conventional and unconventional: dispersion is linear (\"massless'', Dirac-Weyl) in one of the directions of the two-dimensional (2D) layer, and is conventional quadratic (\"massive'' Dirac) in the perpendicular direction. In this dissertation the tightbinding model for the semi-Dirac dispersion is developed. The tightbinding model allows one to describe the semi-Dirac dispersion in terms of an electron hopping from one lattice site to another in a square lattice arrangement. Several low energy features of the semi-Dirac dispersion have been considered. That includes Hall coefficient, magnetic susceptibility and heat capacity. Results for the plasmon dispersion relation in the small momentum regime are obtained and the extreme anisotropy thereof is pointed out. Klein tunneling is considered for the case of normal impingement on a barrier oriented at an arbitrary angle. It is found that perfect transmission occurs under certain conditions. Comparisons of the semi-Dirac system are made throughout with other types of point Fermi surface systems. While for some properties semi-Dirac behavior is intermediate between these two, in some cases it displays several quite distinct properties. The Berry's phase for the semi-Dirac system is computed, and the result obtained is rather unexpected. In the dissertation Quantum Chaos and its relevance to the semi-Dirac dispersion is discussed. The energy level statistics for the semi-Dirac dispersion is obtained, which shows a novel and a very rich behavior compared to other conventional dispersions.

The semi-Dirac semi-Weyl semi-metal has been of interest in recent years due to its naturally occurring point Fermi surface and the associated exotic band-structure near the Fermi surface, which is linear (graphene-like) in one direction of the Brillouin zone, but quadratic in a direction perpendicular to it. In this paper the effect of a magnetic adatom impurity in a semi-Dirac system is studied. As in a metal, the magnetic impurity in a semi-Dirac system interacts with the sea of conduction electrons and gives rise to magnetism. The transition of the semi-Dirac system from the non-magnetic to the magnetic phase is studied as a function of the impurity energy, the strength of hybridization between the impurity and the bath as well as that of the electron electron interaction at the impurity atom. The results are compared and contrasted with those of graphene and ordinary metal. Since the semi-Dirac and the Dirac dispersion share similar features,e.g, both are particle hole symmetric and linear in one direction, the two systems share resemblances in their characteristics in the presence of a magnetic impurity. But some features are unique to the semi-Dirac dispersion.

We extend the study of fermionic particle-hole symmetric semi-Dirac (alternatively, semi-Weyo) dispersion of quasiparticles, \\(\\varepsilon_K = \\pm \\sqrt{(k_x^2/2m)^2 + (vk_y)^2)} = \\pm \\varepsilon_0 \\sqrt{K_x^4 + K_y^2}\\) in dimensionless units, discovered computationally in oxide heterostructures by Pardo and collaborators. This unique system a highly anisotropic sister phase of both (symmetric) graphene and what has become known as a Weyl semimetal, with \\(^{1/2} \\approx v\\) independent of energy, and \\(^{1/2} \\propto m^{-1/2}\\sqrt{\\varepsilon}\\) being very strongly dependent on energy (\\(\\varepsilon\\)) and depending only on the effective mass \\(m\\). Each of these systems is distinguished by bands touching (alternatively, crossing) at a point Fermi surface, with one consequence being that for this semi-Dirac system the ratio \\(|\\chi_{orb}/\\chi_{sp}|\\) of orbital to spin susceptibilities diverges at low doping. We extend the study of the low-energy behavior of the semi-Dirac system, finding the plasmon frequency to be highly anisotropic while the Hall coefficient scales with carrier density in the usual manner. The Faraday rotation behavior is also reported. For Klein tunneling for normal incidence on an arbitrarily oriented barrier, the kinetic energy mixes both linear (massless) and quadratic (massive) contributions depending on orientation. Analogous to graphene, perfect transmission occurs under resonant conditions, except for the specific orientation that eliminates massless dispersion. Comparisons of the semi-Dirac system are made throughout with both other types of point Fermi surface systems.