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Band Structure and Scattering Mechanisms

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Deterministic Solvers for the Boltzmann Transport Equation

Part of the book series: Computational Microelectronics ((COMPUTATIONAL))

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Abstract

The models for the band structure and scattering are discussed in this chapter. The electronic band structure of Si or SiGe is quite complex, and analytical expressions for the dispersion relation involve rather stringent approximations, especially at high energies. On the other hand, direct inclusion of the full band structure into the SHE simulator is at least not possible for the conduction bands. Instead, a valley model of the conduction bands is preferable for SHE, and the development of such a model based on the full band structure is quite important for physically sound SHE simulations. Several band models for electrons, which can be used within the SHE solver, are introduced. In the last section of this chapter, the scattering mechanisms considered in this work are briefly presented.

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Notes

  1. 1.

    In the case of SiGe, the position-dependent band edge of the conduction band due to the position-dependent Ge content is taken into account [16].

  2. 2.

    Before the Herring–Vogt transformation is applied, the DOS is not isotropic.

  3. 3.

    When the device coordinate frame is not aligned with the coordinate system used to perform the SHE of the inverse dispersion relation, a rotation of the spherical harmonics is required. As a result of this rotation more m values can appear for which the fitting coefficients do not vanish. However, since the rotation of the spherical harmonics preserves the original l value, even after an arbitrary rotation, only even l components are required.

  4. 4.

    Note that use of V 1 instead of \(\sqrt{{V }_{2}}\) degrades the quality of the simulation results, as pointed out in [13], because the transport is more closely related with the V 2 moment.

  5. 5.

    Although the derivation of this model is considerably more rigorous than others, there are still missing effects. For example, the antisymmetry against the inversion operation between two valleys of a valley pair is neglected. In principle, this effect can be included at the cost of increased numerical complexity, however, it is neglected in this work. Nevertheless, the simulation result of this model is already quite acceptable.

  6. 6.

    When B E,1, m, 0, 0, d ν coefficients are extracted, unit magnitude of the d-directional force is understood.

  7. 7.

    Since the transverse velocity is much larger than the longitudinal one, the visible difference shown in Fig. 4.8 is counterbalanced by the small difference in Fig. 4.9.

  8. 8.

    The slope of the distribution in the low energy range, calculated by the full band model, is still slightly smaller than that by the FB MC simulation. It seems that the antisymmetry of the valley is the main cause of the remaining discrepancy.

  9. 9.

    For example, at 100 kV cm − 1, the peak of the electron current, which is distributed over the energy, occurs at 0.3 eV.

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Authors and Affiliations

  1. Device Laboratory, Samsung Information Systems America, Inc., 75 West Plumeria Drive, San Jose, CA, 95134, USA

    Sung-Min Hong

  2. Institut für Elektronische Bauelemente und Schaltungstechnik, TU Braunschweig, 3329, 38023, Braunschweig, Germany

    Anh-Tuan Pham

  3. Institut für Theoretische Elektrotechnik, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Christoph Jungemann

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Correspondence to Sung-Min Hong .

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Hong, SM., Pham, AT., Jungemann, C. (2011). Band Structure and Scattering Mechanisms. In: Deterministic Solvers for the Boltzmann Transport Equation. Computational Microelectronics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0778-2_4

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  • DOI: https://doi.org/10.1007/978-3-7091-0778-2_4

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