The Boltzmann Transport Equation and Its Projection onto Spherical Harmonics
In the framework of the semiclassical transport theory, the BTE governs the spatiotemporal evolution of the particle gas. In this chapter, the BTE in the three-dimensional wave vector space is introduced in Sect. 2.1. The PE is required for the calculatio
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The Boltzmann Transport Equation and Its Projection onto Spherical Harmonics
In the framework of the semiclassical transport theory, the BTE governs the spatiotemporal evolution of the particle gas. In this chapter, the BTE in the threedimensional wave vector space is introduced in Sect. 2.1. The PE is required for the calculation of the electric field, which enters the BTE. If only one carrier type is simulated, a drift-diffusion model is solved for the other type. The PE and drift-diffusion model are discussed in Sect. 2.2. In Sect. 2.3, basic properties of the spherical harmonics are reviewed. A generalized coordinate transform from the wavevector space to the energy space is introduced, and some important relations between transport coefficients in the energy space are explicitly derived. The spherical harmonics expansion of the BTE is shown in Sect. 2.5. Finally, noise analysis within the Langevin-Boltzmann framework is discussed in Sect. 2.6.
2.1 The Boltzmann Transport Equation Our goal is to describe the particle kinetics based on a position-dependent band structure, which contains a few conduction (or valence) bands. However, the electronic band, which is defined in the first Brillouin zone following the reduced zone scheme [1], might not be necessarily suitable for a spherical harmonics expansion. For example, in the case of the first Si conduction band, the origin of the Brillouin zone (the -point) does not coincide with the minimum of the energy.1 Therefore, in this work, instead of the electronic bands themselves, a mathematical model called “valley” is regarded as the basis of the analysis. From its construction, the energy minimum of the valley is located at its origin in the wave vector space. The valleys are labeled with throughout part II. Of course, in order to obtain a meaningful physical description, a sound relationship between the original band
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In the case of the Si valence bands, the minimum of the band energy is found at this point.
S.-M. Hong et al., Deterministic Solvers for the Boltzmann Transport Equation, Computational Microelectronics, DOI 10.1007/978-3-7091-0778-2 2, © Springer-Verlag/Wien 2011
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2 The Boltzmann Transport Equation and Its Projection onto Spherical Harmonics
structure and the approximated valley model is required. Specific approaches to construct the band models based on the valley description are explained in Chap. 4. For the th valley, the relative energy at the position r in the real space and the wave vector k, which is measured from the valley minimum energy, is described by the dispersion relation, " .r; k/. Since the band structure can depend on the real space, the dispersion relation of the valley also can be position dependent. In the framework of the semiclassical transport theory, the position r in the real space and the momentum „k („ denotes Planck’s constant divided by 2) of a particle can be measured simultaneously. Therefore, a state .r; k/ in the sixdimensional phase space is required in order to specify the state of a particle.
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