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 EIT4

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

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

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

Theory

Langevin-Boltzmann equation: Projection onto spherical harmonics Y l,m : Expansion on equienergy surfaces -Simpler expansion -Energy conservation (magnetic field, scattering) -FB compatible Angles are the same as in k-space New variables: (  unique inversion required) Delta function leads to generalized DOS

Theory Generalized DOS (d 3 k  d  d  ): 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

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)

FB bulk results for holes

Heavy hole band of silicon (k z =0, l max =20) g, E=30kV/cm in [110]DOS

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

1D NPN BJT

V CE =0.5V SHE can handle small currents without problems 50nm NPN BJT Modena model for electrons with analytical band structure

1D NPN BJT V CE =0.5V SHE can handle huge variations in the density without problems V CE =0.5V, V BE =0.55V

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

1D NPN BJT A 2nm grid spacing seems to be sufficient V CE =0.5V, V BE =0.85V Dependence on grid spacing

1D NPN BJT Rapidly varying electric fields pose no problem Grid spacing varies from 1 to 10nm V CE =3.0V, V BE =0.85V

1D NPN BJT V CE =1.0V, V BE =0.85V

1D NPN BJT Collector current noise, V CE =0.5V, f=0Hz Up to high injection the noise is shot-like (S CC =2qI C )

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

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.

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