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Microstructural Characterization of Conjugated Organic Semiconductors by X-Ray Scattering
Published in John R. Reynolds, Barry C. Thompson, Terje A. Skotheim, Conjugated Polymers, 2019
Maged Abdelsamie, Michael F. Toney
One important concept that correlates the crystal structure to the electronic structure of semiconductors is the Brillouin zone, introduced by L. Brillouin [34]. Brillouin zone is a unit cell of the reciprocal lattice that is enclosed by the Bragg planes and can be constructed by the perpendicular bisectors of the reciprocal lattice vectors drawn from the origin of reciprocal space, see Figure 12.2C. The smallest volume enclosed by Bragg planes constitutes the 1st Brillouin zone, as shown for 2D space in Figure 12.2C. The higher order Brillouin zones can be constructed using the higher order Bragg planes. Through density functional theory (DFT) calculations, the electronic band structure can be resolved. Characterization of the crystal structure (and molecular packing) is crucial for understanding the electronic structure for organic semiconductors (OSCs) and has been implemented successfully in the electronic band structure calculations for OSCs [35, 36].
Some physical properties of silicon
Published in O.A. Aktsipetrov, I.M. Baranova, K.N. Evtyukhov, Second Order Non-linear Optics of Silicon and Silicon Nanostructures, 2018
O.A. Aktsipetrov, I.M. Baranova, K.N. Evtyukhov
Usually we consider the so-called first Brillouin zone, for which the range of variation of the vector k is given by –π ≤ k · ai≤ π (i = 1, 2, 3). Such Brillouin zone is a parallelepiped in k-space. The volume of the first Brillouin zone in k-space is equal to the volume of the unit cell of the reciprocal lattice VB (see formula (1.8)). Obviously, the volume of the Brillouin zone is measured in m–3.
Introductory Concepts
Published in Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck, Computational Electronics, 2017
Dragica Vasileska, Stephen M. Goodnick, Gerhard Klimeck
The Brillouin zone [7] is a primitive unit cell of the reciprocal lattice. In mathematics and solid-state physics, the first Brillouin zone is a uniquely defined primitive cell of the reciprocal lattice in the frequency domain. It is found by the same method as for the Wigner–Seitz cell in the Bravais lattice. The importance of the Brillouin zone stems from the Bloch wave description of waves in a periodic medium, in which it is found that the solutions can be completely characterized by their behavior in a single Brillouin zone. Taking the surfaces at the same distance from one element of the lattice and its neighbors, the volume included is the first Brillouin zone shown in Figure 2.9. Another definition is as the set of points in k-space that can be reached from the origin without crossing any Bragg plane. There are also second, third, etc. Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used more rarely. As a result, the first Brillouin zone is often called simply the Brillouin zone. (In general, the nth Brillouin zone consists of the set of points that can be reached from the origin by crossing n − 1 Bragg planes.)
A convergent low-wavenumber, high-frequency homogenization of the wave equation in periodic media with a source term
Published in Applicable Analysis, 2022
Shixu Meng, Othman Oudghiri-Idrissi, Bojan B. Guzina
The key elements of (Floquet-)Bloch transform introduced in the sequel are adapted from [31] (see also [5]). To begin with, we introduce the Wigner-Seitz cell This gives the dual or reciprocal lattice and the Brillouin zone With reference to the periodicity and arbitrary wavenumber , any function that satisfies is called -quasiperiodic with respect to . In passing, we note that any -periodic function becomes -quasiperiodic upon multiplication by . Note that if a function is -quasiperiodic with respect to , then its values (in the variable ) at fully determines its values at .
Stability of X-IV-IV half Heusler semiconductor alloys: a DFT study
Published in Molecular Physics, 2021
We consider the unit cell of the alloy containing three atoms in this investigation. To promote convergence, we built a supercell. The computations occur in the first Brillouin zone, representing the primitive cell (not conventional cell) in the reciprocal lattice. The zones sampled in this work covers Γ – X – K – Γ- L – W – X. the three alloys prove to be dynamically stable; this is evident in the fact that from Figure 3 (a-e), there are no negative frequencies or dispersions. Considering that there are three atoms, using the relation 3n dispersion branches, where n is the number of atoms, we observe nine modes in the phonon dispersion spectra in Figs. (b, d, f); there are three acoustic modes (3) and six optical modes (3n-3). The optical modes can interact with light. The optical mode for NiHfSi is between 250–330 cm−1; this puts the wavelength of the alloy between 30303 nm and 40000 nm in the Far-IR wavelength region. The frequency range for the optical mode in NiHfSn and NiHfGe is between 160 −240 cm−1 and 180 −230 cm−1. The alloys are all in the far-infrared region, and a much higher frequency will be required to bring it to the visible region, which is most desirable in solar systems.
Computational study of seismic wave propagation through metamaterial foundation
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2021
The first Brillouin zone [5] for a unit cell with assumed configurations is expressed by the pyramid as shown in Figure 4, where a is the length of unit cell. Once the attenuation zones or theoretical frequency bandgap have been computed, the next stage is to see the response reduction of these periodic foundations when exposed to various excitations and to check its attenuation efficiency in frequency bandgap zones. This can be achieved by plotting the FRF, which can be obtained using Equation (6), subjected to transverse wave (S-waves) and longitudinal waves (P-waves). is the input unit amplitude displacement provided usually at the base of periodic panel and is the displacement measured at the top surface.