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Density Matrix
Published in David K. Ferry, An Introduction to Quantum Transport in Semiconductors, 2017
In computing an estimate of the size of the electron’s wave packet, we follow an approach similar to that of Wannier functions in determining impurity ionization energies. A Wannier function is a localized wavefunction that is obtained by summing over all Bloch functions; the Bloch functions are taken from a single band in the effective mass approximation [70]. This approach finds its most notable use in determining the wavefunction for an electron trapped on an impurity. In this latter case, the wavefunction radius is found to be essentially the Bohr radius in the semiconductor (about 3.3 nm in Si). This wavefunction, in real space, is the “amplitude” function describing the expansion in Wannier functions and is termed the envelope function. We follow this procedure by describing the occupancy of the Bloch functions by the appropriate distribution function and then computing the proper envelope function.
Introductory Concepts
Published in Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck, Computational Electronics, 2017
Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck
in which unk is the periodic part of the Bloch wave and the integral is over the Brillouin zone. Here, index n refers to the nth energy band in the crystal. The Wannier functions are localized near atomic sites, like atomic orbitals, but being defined in terms of Bloch functions, they are accurately related to solutions based upon the crystal potential. Wannier functions on different atomic sites R are orthogonal. The Wannier functions can be used to form the Schrödinger solution for the nth energy band as
Representation of the virtual space in extended systems – a correlation energy convergence study
Published in Molecular Physics, 2020
A. S. Hansen, G. Baardsen, E. Rebolini, L. Maschio, T. B. Pedersen
The arguments underpinning linear-scaling correlation treatments thus rely heavily on the concept of orbital locality. As mentioned in the Introduction, this concept is not uniquely defined and a number of localisation functionals have been proposed. In this work, we will consider the central-moment functionals of Høyvik and Jørgensenalong with their statistics-based measures of orbital locality [4]. The mth power of the second central moment (PSM-m) functionals are defined in terms of the second moment orbital spread of each orbital p, as where the summation over orbitals should be restricted to either occupied or virtual orbitals to maintain the Brillouin condition. The PSM-1 functional is identical to the Foster-Boys functional [36, 37]. Minimising the PSM-1 functional with respect to unitary rotations of the (orthonormal) orbitals leads to the set of orbitals with the smallest possible sum of orbital spreads. In the context of periodic systems, such orbitals are commonly referred to as maximally localised Wannier functions [38].