C. Jungemann Institute for Electronics University of the Armed Forces Munich, Germany
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Transcript of C. Jungemann Institute for Electronics University of the Armed Forces Munich, Germany
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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
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Outline
• Introduction
• Theory
• FB bulk results for holes
• Results for a 1D NPN BJT
• Conclusions
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Introduction
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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
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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
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Theory
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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
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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
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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)
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FB bulk results for holes
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FB bulk results for holes
Heavy hole band of silicon (kz=0, lmax=20)
g, E=30kV/cm in [110]DOS
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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
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1D NPN BJT
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1D NPN BJT
VCE=0.5V
SHE can handle small currents without problems
50nm NPN BJT
Modena model for electronswith analytical band structure
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1D NPN BJT
VCE=0.5V
SHE can handle huge variations in the density without problems
VCE=0.5V, VBE=0.55V
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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
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1D NPN BJT
A 2nm grid spacing seems to be sufficient
VCE=0.5V, VBE=0.85V
Dependence on grid spacing
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1D NPN BJT
Rapidly varying electric fields pose no problemGrid spacing varies from 1 to 10nm
VCE=3.0V, VBE=0.85V
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1D NPN BJT
VCE=1.0V, VBE=0.85V
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1D NPN BJT
Collector current noise, VCE=0.5V, f=0Hz
Up to high injection the noise is shot-like (SCC=2qIC)
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1D NPN BJT
Collector current noise, VCE=0.5V, f=0Hz
Spatial origin of noise can not be determined by MC
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Conclusions
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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.
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Conclusions
• Self consistent solution of BE and PE with a full Newton
• AC analysis possible (at arbitrary frequencies)
• Noise analysis possible