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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. - PowerPoint PPT Presentation
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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