Abstract
An analytical formula is derived for particle and energy densities of fermions and bosons, and their ballistic momentum and energy currents for anisotropic energy dispersions in generalized dimensions. The formulation considerably simplifies the comparison of the statistical properties and ballistic particle and energy transport currents of electrons, acoustic phonons, and photons in various dimensions in a unified manner. Assorted examples of its utility are discussed, ranging from blackbody radiation to Schottky diodes and ballistic transistors, quantized electrical and thermal conductance, generalized ballistic Seebeck and Peltier coefficients, their Onsager relations, the generalized Wiedemann-Franz law and the robustness of the Lorenz number, and ballistic thermoelectric power factors, all of which are obtained from the single formula. The new formulation predicts a thermoelectric power factor behaviour of 3D Dirac bands which has not been observed yet.
Highlights
The need for analytical expressions for particle, energy, and current densities arises frequently in various branches of science and engineering
The generalized ballistic current expression obtained in Eq (4) is found to be a versatile tool to compute and compare in a unified manner the particle and energy densities, charge and energy currents, thermoelectric coefficients, and more for fermions and bosons of various energy dispersions
Such a compact formulation is well suited for optimization problems, in which the extrema of one or more densities, currents, transport coefficients, or their combinations need to be determined as a function of the dimensionality, type of dispersion, effective masses, wave velocities, etc
Summary
The need for analytical expressions for particle, energy, and current densities arises frequently in various branches of science and engineering. They are typically handled separately for each case of interest. A single unified analytical expression is found in this work for all the above densities and their ballistic momentum and energy currents for anisotropic dispersions in the noninteracting ballistic transport regime. This enables particle and energy densities, and ballistic particle and energy transport currents of electrons, phonons, and photons to be treated in a unified manner amplifying their similarities and differences, the need for which has been advocated [1]. The parameters β and μ in f± are dictated by the respective reservoirs, and the group velocity neglects Berryphase contributions
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