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C. Jungemann Institute for Electronics University of the Armed Forces Munich, Germany

EIT4. Stable Discretization of the Langevin-Boltzmann equation based on Spherical Harmonics, Box Integration, and a Maximum Entropy Dissipation Scheme. C. Jungemann Institute for Electronics University of the Armed Forces Munich, Germany.

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C. Jungemann Institute for Electronics University of the Armed Forces Munich, Germany

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  1. EIT4 Stable Discretization of the Langevin-Boltzmann equation based on Spherical Harmonics, Box Integration, and a Maximum Entropy Dissipation Scheme C. Jungemann Institute for Electronics University of the Armed Forces Munich, Germany Acknowledgements: C. Ringhofer, M. Bollhöfer, A. T. Pham, B. Meinerzhagen

  2. Outline • Introduction • Theory • FB bulk results for holes • Results for a 1D NPN BJT • Conclusions

  3. Introduction

  4. Introduction • Macroscopic models fail for strong nonequilibrium • Macroscopic models also fail near equilibrium in nanometric devices • Full solution of the BE is required • MC has many disadvantages (small currents, frequencies below 100GHz, ac) 1D 40nm N+NN+ structure

  5. Introduction A deterministic solver for the BE is required Main objectives: • SHE of arbitrary order for arbitrary band structures including full band and devices • Exact current continuity without introducing it as an additional constrain • Stabilization without relying on the H-transform • Self consistent solution of BE and PE • Stationary solutions, ac and noise analysis

  6. Theory

  7. Theory Langevin-Boltzmann equation: Projection onto spherical harmonics Yl,m: • Expansion on equienergy surfaces • Simpler expansion • Energy conservation (magnetic field, scattering) • FB compatible • Angles are the same as in k-space • New variables: (e,J,j) (unique inversion required) • Delta function leads to generalized DOS

  8. Theory Generalized DOS (d3kgdedW): Generalized energy distribution function: The particle density is given by: With g the drift term can be expressed with a 4D divergence and box integration results in exact current continuity

  9. Theory • Stabilization is achieved by application of a maximum entropy dissipation principle(see talk by C. Ringhofer) • Due to linear interpolation of the quasistatic potential this corresponds to a generalized Scharfetter-Gummel scheme • BE and PE solved with the Newton method • Resultant large system of equations is solved CPU and memory efficiently with the robust ILUPACK solver (see talk by M. Bollhöfer)

  10. FB bulk results for holes

  11. FB bulk results for holes Heavy hole band of silicon (kz=0, lmax=20) DOS g, E=30kV/cm in [110]

  12. FB bulk results for holes Holes in silicon (lmax=13) Drift velocity g0,0, E in [110] SHE can handle anisotropic full band structures and is not inferior to MC

  13. 1D NPN BJT

  14. 1D NPN BJT Modena model for electronswith analytical band structure VCE=0.5V 50nm NPN BJT SHE can handle small currents without problems

  15. 1D NPN BJT VCE=0.5V, VBE=0.55V VCE=0.5V SHE can handle huge variations in the density without problems

  16. 1D NPN BJT Dependence on the maximum order of SHE VCE=0.5V, VBE=0.85V Transport in nanometric devices requires at least 5th order SHE

  17. 1D NPN BJT Dependence on grid spacing VCE=0.5V, VBE=0.85V A 2nm grid spacing seems to be sufficient

  18. 1D NPN BJT VCE=3.0V, VBE=0.85V Rapidly varying electric fields pose no problem Grid spacing varies from 1 to 10nm

  19. 1D NPN BJT VCE=1.0V, VBE=0.85V

  20. 1D NPN BJT Collector current noise, VCE=0.5V, f=0Hz Up to high injection the noise is shot-like (SCC=2qIC)

  21. 1D NPN BJT Collector current noise, VCE=0.5V, f=0Hz Spatial origin of noise can not be determined by MC

  22. Conclusions

  23. Conclusions • SHE is possible for FB. At least if the energy wave vector relation can be inverted. • Exact current continuity by virtue of construction due to box integration and multiplication with the generalized DOS. • Robustness of the discretization based on the maximum entropy dissipation principle is similar to macroscopic models. • Convergence of SHE demonstrated for nanometric devices.

  24. Conclusions • Self consistent solution of BE and PE with a full Newton • AC analysis possible (at arbitrary frequencies) • Noise analysis possible

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