Abstract
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 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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
In the case of the Si valence bands, the minimum of the band energy is found at this point.
- 2.
In the literature, the order l and the sub-order m are usually called the degree l and the order m, respectively.
- 3.
Since the coordinate transform is applied to the six-dimensional phase space, this is only a part of the total Jacobian matrix, corresponding to the k space. Because the transform for the real space is just an identity transform, the entire 6 ×6 Jacobian matrix can be inverted blockwise.
- 4.
This definition does not include the integration over the solid angle. The generalized DOS is therefore for the case that it does not depend on the angles by a factor of 4π smaller than the conventional expression.
- 5.
The polar optical phonon, which is important for III–V materials, cannot be properly treated as a velocity-randomizing one. In this case, higher-order expansions of the transition rate should be considered, as shown in [15].
- 6.
Noise analysis under large-signal conditions has been demonstrated for bulk simulations within the framework of the deterministic BTE solver in [17].
- 7.
From this definition, the expectation of a fluctuation is always zero.
- 8.
A formulation which keeps the spin variable can be found in [21].
References
Madelung, O.: Introduction to Solid State Theory. Springer, Berlin (1978)
van Kampen, N.G.: Stochastic Process in Physics and Chemistry. North-Holland, Amsterdam (1981)
Price, P.J.: Monte Carlo calculation of electron transport in solids. Semiconduct. Semimet. 14, 249–309 (1979)
Jacoboni, C., Lugli, P.: The Monte Carlo Method for Semiconductor Device Simulation. Springer, New York (1989)
Hong, S.-M., Jungemann, C.: A fully coupled scheme for a Boltzmann-Poisson equation solver based on a spherical harmonics expansion. J. Comput. Electron 8(3), 225–241 (2009)
Hong, S.-M., Matz, G., Jungemann, C.: A deterministic Boltzmann equation solver based on a higher-order spherical harmonics expansion with full-band effects. IEEE Trans. Electron Dev. 57, 2390–2397 (2010)
Jungemann, C., Meinerzhagen, B.: Hierarchical Device Simulation: The Monte-Carlo Perspective, Computational Microelectronics. Springer, New York (2003)
Selberherr, S.: Analysis and Simulation of Semiconductor Devices. Springer, New York (1984)
Banoo, K., Lundstrom, M.: Direct solution of the Boltzmann transport equation in nanoscale Si devices. In: Proceedings of SISPAD, pp. 50–53 (2000)
Lisle, I.G., Huang, S.-L.T.: Algorithms for spherical harmonic lighting. In: GRAPHITE ’07: Proceedings of the 5th international conference on Computer graphics and interactive techniques in Australia and Southeast Asia, pp. 235–238. ACM, New York (2007)
Jungemann, C., Pham, A.-T., Meinerzhagen, B., Ringhofer, C., Bollhöfer, M.: Stable discretization of the Boltzmann equation based on spherical harmonics, box integration, and a maximum entropy dissipation principle. J. Appl. Phys. 100, 024502–1–13 (2006)
Gnudi, A., Ventura, D., Baccarani, G., Odeh, F.: Two-dimensional MOSFET simulation by means of a multidimensional spherical harmonics expansion of the Boltzmann transport equation. Solid State Electron 36(4), 575–581 (1993)
Liang, W., Goldsman, N., Mayergoyz, I., Oldiges, P.J.: 2- D MOSFET modeling including surface effects and impact ionization by self-consistent solution of the Boltzmann, Poisson, and hole-continuity equations. IEEE Trans. Electron Dev. 44(2), 257–267 (1997)
Kosina, H.: A method to reduce small-angle scattering in Monte Carlo device analysis. IEEE Trans. Electron Dev. 46(6), 1196–1200 (1999)
Bieder, J., Hong, S.-M., Jungemann, C.: A deterministic Boltzmann solver for GaAs devices based on the spherical harmonics expansion. In: Proceedings of IWCE International Workshop on Computational Electronics, pp. 31–34 (2010)
Bonani, F., Ghione, G.: Noise in Semiconductor Devices, Modeling and Simulation, Advanced Microelectronics. Springer, Heidelberg (2001)
Jungemann, C., Meinerzhagen, B.: A frequency domain spherical harmonics solver for the Langevin Boltzmann equation. In: International Conference on Noise in Physical Systems and 1/f Fluctuations, AIP Conference Proceedings vol. 780, pp. 777–782 (2005)
Gardiner, C.W.: Handbook of Stochastic Methods. Springer, Berlin (1985)
Kogan, Sh.: Electronic Noise and Fluctuations in Solids. Cambridge University Press, Cambridge (1996)
Papoulis, A.: Probability, Random Variables, and Stochastic Processes, 3rd edn. McGraw–Hill, NY (1991)
Jungemann, C.: A deterministic solver for the Langevin Boltzmann equation including the Pauli principle. In: SPIE: Fluctuations and Noise, vol. 6600, pp. 660007–1–660007–12 (2007)
Jackson, J.D.: Classical Electrodynamics, 2nd edn. Wiley, New York (1975)
Gantsevich, S.V., Gurevich, V.L., Katilius, R.: Theory of fluctuations in nonequilibrium electron gas. Nuovo Cimento 2, 5 (1979)
Branin, F.H.: Network sensitivity and noise analysis simplified. IEEE Trans. Circuit Theor. 20, 285–288 (1973)
Bonani, F., Ghione, G., Pinto, M.R., Smith, R.K.: An efficient approach to noise analysis through multidimentional physics-based models. IEEE Trans. Electron Dev. 45(1), 261–269 (1998)
Jungemann, C.: Transport and noise calculations for nanoscale Si devices based on the Langevin Boltzmann equation expanded with spherical harmonics. J. Comput. Theor. Nanosci. 5(6), 1152–1169 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2011 Springer-Verlag/Wien
About this chapter
Cite this chapter
Hong, SM., Pham, AT., Jungemann, C. (2011). The Boltzmann Transport Equation and Its Projection onto Spherical Harmonics. In: Deterministic Solvers for the Boltzmann Transport Equation. Computational Microelectronics. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0778-2_2
Download citation
DOI: https://doi.org/10.1007/978-3-7091-0778-2_2
Published:
Publisher Name: Springer, Vienna
Print ISBN: 978-3-7091-0777-5
Online ISBN: 978-3-7091-0778-2
eBook Packages: EngineeringEngineering (R0)