Stable Discretization of the Langevin-Boltzmann equation based on Spherical

Harmonics, Box Integration, and a Maximum Entropy Dissipation Scheme

C. Jungemann

Institute for ElectronicsUniversity 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

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

Theory

Langevin-Boltzmann equation:

fSfht

f ˆ,

Projection onto spherical harmonics Yl,m:

kdfSfh

t

fYk ml

3,3

ˆ,),()(2

2

•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

),,( with )2(

),,(3

2

kkkk

Z

)),,,(,(),,(2),,,,( tkrfZtrg

Generalized DOS (d3kdd):

Generalized energy distribution function:

The particle density is given by:

dtrgY

trn ),,(1

),( 0,00,0

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

FB bulk results for holes

Heavy hole band of silicon (kz=0, lmax=20)

g, E=30kV/cm in [110]DOS

FB bulk results for holes

Holes in silicon (lmax=13)

g0,0, E in [110]Drift velocity

SHE can handle anisotropic full band structures and is not inferior to MC

1D NPN BJT

1D NPN BJT

VCE=0.5V

SHE can handle small currents without problems

50nm NPN BJT

Modena model for electronswith analytical band structure

1D NPN BJT

VCE=0.5V

SHE can handle huge variations in the density without problems

VCE=0.5V, VBE=0.55V

1D NPN BJT

Transport in nanometric devices requires at least 5th order SHE

VCE=0.5V, VBE=0.85V

Dependence on the maximum order of SHE

1D NPN BJT

A 2nm grid spacing seems to be sufficient

VCE=0.5V, VBE=0.85V

Dependence on grid spacing

1D NPN BJT

Rapidly varying electric fields pose no problemGrid spacing varies from 1 to 10nm

VCE=3.0V, VBE=0.85V

1D NPN BJT

VCE=1.0V, VBE=0.85V

1D NPN BJT

Collector current noise, VCE=0.5V, f=0Hz

Up to high injection the noise is shot-like (SCC=2qIC)

1D NPN BJT

Collector current noise, VCE=0.5V, f=0Hz

Spatial origin of noise can not be determined by MC

Conclusions

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