EIT4
Stable Discretization of the LangevinBoltzmann 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
Outline
•
•
•
•
•
Introduction
Theory
FB bulk results for holes
Results for a 1D NPN BJT
Conclusions
Introduction
Introduction
1D 40nm N+NN+ structure
• 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)
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:
f
 h, f   Sˆ f  
t
Projection onto spherical harmonics Yl,m:
2
2 
3



 f
 3
ˆ
     (k ) Yl ,m (, ) t  h, f   S f   d k

•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 (d3kgddW):
k k

Z ( , ,  ) 
with k  k ( , ,  )
3
(2 ) 
2

Generalized energy distribution function:



 
g (r ,  , ,  , t )  2Z ( , ,  ) f (r , k ( , ,  ), t )

The particle density is given by:

1
n(r , t ) 
Y0,0

  g0,0 (r ,  , t )d


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)
DOS
g, E=30kV/cm in [110]
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
1D NPN BJT
1D NPN BJT
Modena model for electrons
with analytical band structure
50nm NPN BJT
VCE=0.5V
SHE can handle small currents without problems
1D NPN BJT
VCE=0.5V, VBE=0.55V
VCE=0.5V
SHE can handle huge variations in the density without problems
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
1D NPN BJT
Dependence on grid spacing
VCE=0.5V, VBE=0.85V
A 2nm grid spacing seems to be sufficient
1D NPN BJT
VCE=3.0V, VBE=0.85V
Rapidly varying electric fields pose no problem
Grid spacing varies from 1 to 10nm
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