Abstract
In this chapter we discuss how to treat electronic transport in small structures with a full quantum-mechanical formulation; that is, we introduce the problem of “quantum transport.” We introduce the concept of Green’s functions and see how they are related to the density matrix. In order to find a solution of the transport problem, we consider first the case of particles in the presence of a spatially dependent potential, but not interacting with the excitations of the lattice, impurities, or among themselves. This constitutes “ballistic transport” and we see how to formulate it as a Schrödinger equation with open boundary conditions. We discuss the Quantum Transmitting Boundary Method that can be used to solve this problem. Numerical procedures are also presented in some detail. We finally consider the case of many particles interacting with lattice excitations and other perturbations. We outline various methods of solution: Those based on Green’s functions, namely, the Non-Equilibrium Green’s Function formulation and those based on the equation of motion for the density matrix, leading to various Master equations, to the Wigner function formulation, and to Semiconductor Bloch Equations. We briefly discuss the Wigner function approach, since the density-matrix approach will be treated in detail in the following chapter.
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Fischetti, M.V., Vandenberghe, W.G. (2016). Overview of Quantum-Transport Formalisms. In: Advanced Physics of Electron Transport in Semiconductors and Nanostructures. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01101-1_17
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DOI: https://doi.org/10.1007/978-3-319-01101-1_17
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