1. Semi-Classical Transport Theory

2. Outline: What is Computational Electronics?
Semi-Classical Transport Theory
Drift-Diffusion Simulations
Hydrodynamic Simulations
Particle-Based Device Simulations
Inclusion of Tunneling and Size-Quantization Effects in Semi-Classical Simulators
Tunneling Effect: WKB Approximation and Transfer Matrix Approach
Quantum-Mechanical Size Quantization Effect
Drift-Diffusion and Hydrodynamics: Quantum Correction and Quantum Moment Methods
Particle-Based Device Simulations: Effective Potential Approach
Quantum Transport
Direct Solution of the Schrodinger Equation (Usuki Method) and Theoretical Basis of the Green’s Functions Approach (NEGF)
NEGF: Recursive Green’s Function Technique and CBR Approach
Atomistic Simulations – The Future
Prologue

3. Semi-Classical Transport Theory
It is based on direct or approximate solution of the Boltzmann Transport Equation for the semi-classical distribution function f(r,k,t)
which gives one the probability of finding a particle in region (r,r+dr) and (k,k+dk) at time t
Moments of the distribution function give us information about:
Particle Density
Current Density
Energy Density
D. K. Ferry, Semiconductors, MacMillian, 1990.

4. Semi-Classical Transport Approaches
1. Drift-Diffusion Method
2. Hydrodynamic Method
3. Direct Solution of the Boltzmann Transport Equation via:
Particle-Based Approaches – Monte Carlo method
Spherical Harmonics
Numerical Solution of the Boltzmann-Poisson Problem
C. Jacoboni, P. Lugli: "The Monte Carlo Method for Semiconductor Device Simulation“, in series "Computational Microelectronics", series editor: S. Selberherr; Springer, 1989, ISBN:

5. 1. Drift-Diffusion Approach
Constitutive Equations
Poisson
Continuity Equations
Current Density Equations
S. Selberherr: "Analysis and Simulation of Semiconductor Devices“, Springer, 1984.

6. Numerical Solution Details
Linearization of the Poisson equation
Scharfetter-Gummel Discretization of the Continuity equation
De Mari scaling of variables
Discretization of the equations
Finite Difference – easier to implement but requires more node points, difficult to deal with curved interfaces
Finite Elements – standard, smaller number of node points, resolves curved surfaces
Finite Volume

7. Linearized Poisson Equation φ→φ + δ where δ= φnew - φold
Finite difference discretization:
Potential varies linearly between mesh points
Electric field is constant between mesh points
Linearization → Diagonally-dominant coefficient matrix A is obtained

8. Scharfetter-Gummel Discretization of the Continuity Equation
Electron and hole densities n and p vary exponentially between mesh points → relaxes the requirement of using very small mesh sizes
The exponential dependence of n and p upon the potential is buried in the Bernoulli functions
Bernoulli function:

9. Scaling due to de Mari

10. Numerical Solution Details
Governing Equations ICS/BCS
System of Algebraic Equations
Equation (Matrix) Solver
ApproximateSolution
Discretization
φi (x,y,z,t)
p (x,y,z,t)
n (x,y,z,t)
Continuous Solutions
Finite-Difference
Finite-Volume
Finite-Element
Spectral
Boundary Element
Hybrid
Discrete Nodal Values
Tridiagonal
SOR
Gauss-Seidel
Krylov
Multigrid
D. Vasileska, EEE533 Semiconductor Device and Process Simulation Lecture Notes,
Arizona State University, Tempe, AZ.

11. Numerical Solution Details
Poisson solvers:
Direct
Gaussian Eliminatioln
LU decomposition
Iterative
Mesh Relaxation Methods
Jacobi, Gauss-Seidel, Successive over-Relaxation
Advanced Iterative Solvers
ILU, Stone’s strongly implicit method, Conjugate gradient methods and Multigrid methods
G. Speyer, D. Vasileska and S. M. Goodnick, "Efficient Poisson solver for semiconductor device modeling using the multi-grid preconditioned BICGSTAB method", Journal of Computational Electronics, Vol. 1, pp (2002).

12. Complexity of linear solvers
Time to solve model problem (Poisson’s equation) on regular mesh
n1/2 2D 3D
Sparse Cholesky:
O(n1.5 )
O(n2 )
CG, exact arithmetic:
CG, no precond:
O(n1.33 )
CG, modified IC:
O(n1.25 )
O(n1.17 )
CG, support trees:
O(n1.20 ) -> O(n1+ )
O(n1.75 ) -> O(n1.31 )
Multigrid:
O(n)

13. LU Decomposition

14. Numerical Solution Details of the Coupled Equation Set
Solution Procedures:
Gummel’s Approach – when the constitutive equations are weekly coupled
Newton’s Method – when the constitutive equations are strongly coupled
Gummel/Newton – more efficient approach
D. Vasileska and S. M. Goodnick, Computational Electronics, Morgan & Claypool,
2006.
D. Vasileska, S. M. Goodnick and G. Klimeck, Computational Electronics: From
Semiclassical to Quantum Transport Modeling, Taylor & Francis, in press.

15. Schematic Description of the Gummel’s Approach

16. Constraints on the MESH Size and TIME Step
The time step and the mesh size may correlate to each other in connection with the numerical stability:
The time step t must be related to the plasma frequency
The Mesh size is related to the Debye length

17. When is the Drift-Diffusion Model Valid?
Large devices where the velocity of the carriers is smaller than the saturation velocity
The validity of the method can be extended for velocity saturated devices as well by introduction of electric field dependent mobility and diffusion coefficient:
Dn(E) and μn(E)
25 nm
100 nm
140 nm

18. What Contributes to The Mobility?
D. Vasileska and S. M. Goodnick, "Computational Electronics", Materials Science and Engineering, Reports: A Review Journal, Vol. R38, No. 5, pp (2002)

19. Mobility Modeling Mobility modeling can be separated in three parts:
Low-field mobility characterization for bulk or inversion layers
High-field mobility characterization to account for velocity saturation effect
Smooth interpolation between the low-field and high-field regions
Silvaco ATLAS Manual.

20. (A) Low-Field Models for Bulk Materials
Phonon scattering:
- Simple power-law dependence of the temperature
- Sah et al. model: acoustic + optical and intervalley phonons combined via Mathiessen’s rule
Ionized impurity scattering:
- Conwell-Weiskopf model
- Brooks-Herring model

21. (A) Low-Field Models for Bulk Materials (cont’d)
Combined phonon and ionized impurity scattering:
- Dorkel and Leturg model:
temperature-dependent phonon scattering + ionized impurity scattering + carrier-carrier interactions
- Caughey and Thomas model:
temperature independent phonon scattering + ionized impurity scattering

22. (A) Low-Field Models for Bulk Materials (cont’d)
- Sharfetter-Gummel model:
phonon scattering + ionized impurity scattering (parameterized expression – does not use the Mathiessen’s rule)
- Arora model:
similar to Caughey and Thomas, but with temperature dependent phonon scattering

23. (A) Low-Field Models for Bulk Materials (cont’d)
Carrier-carrier scattering
- modified Dorkel and Leturg model
Neutral impurity scattering:
- Li and Thurber model:
mobility component due to neutral impurity scattering is combined with the mobility due to lattice, ionized impurity and carrier-carrier scattering via the Mathiessen’s rule

24. (B) Field-Dependent Mobility

25. (C) Inversion Layer Mobility Models
CVT model:
combines acoustic phonon, non-polar optical phonon and surface-roughness scattering (as an inverse square dependence of the perpendicular electric field) via Mathiessen’s rule
Yamaguchi model:
low-field part combines lattice, ionized impurity and surface-roughness scattering
there is also a parametric dependence on the in-plane field (high-field component)

26. (C) Inversion Layer Mobility Models (cont’d)
Shirahata model:
uses Klaassen’s low-field mobility model
takes into account screening effects into the inversion layer
has improved perpendicular field dependence for thin gate oxides
Tasch model:
the best model for modeling the mobility in MOS inversion layers; uses universal mobility behavior

27. Generation-Recombination Mechanisms Classification

28. Hydrodynamic Modeling
In small devices there exists non-stationary transport and carriers are moving through the device with velocity larger than the saturation velocity
In Si devices non-stationary transport occurs because of the different order of magnitude of the carrier momentum and energy relaxation times
In GaAs devices velocity overshoot occurs due to intervalley transfer
T. Grasser (ed.): "Advanced Device Modeling and Simulation“, World Scientific Publishing Co., 2003, ISBN: M.
M. Lundstrom, Fundamentals of Carrier Transport, 1990.

29. Velocity Overshoot in Silicon
Scattering mechanisms:
Acoustic deformation potential scattering
Zero-order intervalley scattering (f and g-phonons)
First-order intervalley scattering (f and g-phonons)
X. He, MS Thesis, ASU, 2000.

30. How is the Velocity Overshoot Accounted For?
In Hydrodynamic/Energy balance modeling the velocity overshoot effect is accounted for through the addition of an energy conservation equation in addition to:
Particle Conservation (Continuity Equation)
Momentum (mass) Conservation Equation

31. Hydrodynamic Model due to Blotakjer
Constitutive Equations: Poisson +

32. Closure

33. Momentum Relaxation Rate
K. Tomizawa, Numerical Simulation Of Submicron Semiconductor Devices.

34. Energy Relaxation Rate
K. Tomizawa, Numerical Simulation Of Submicron Semiconductor Devices.

35. Validity of the Hydrodynamic Model
90 nm device
SR = series resistance
Silvaco ATLAS simulations performed by Prof. Vasileska.

36. Validity of the Hydrodynamic Model
25 nm device
SR = series resistance
Silvaco ATLAS simulations performed by Prof. Vasileska.

37. Validity of the Hydrodynamic Model
14 nm device
SR = series resistance
Silvaco ATLAS simulations performed by Prof. Vasileska.

38. Failure of the Hydrodynamic Model
14 nm
25 nm
90 nm
Silvaco ATLAS simulations performed by Prof. Vasileska.

39. Failure of the Hydrodynamic Model
14 nm
25 nm
90 nm
Silvaco ATLAS simulations performed by Prof. Vasileska.

40. Failure of the Hydrodynamic Model
14 nm
25 nm
90 nm
Silvaco ATLAS simulations performed by Prof. Vasileska.

41. Summary
Drift-Diffusion model is good for large MOSFET devices, BJTs, Solar Cells and/or high frequency/high power devices that operate in the velocity saturation regime
Hydrodynamic model must be used with caution when modeling devices in which velocity overshoot, which is a signature of non-stationary transport, exists in the device
Proper choice of the energy relaxation times is a problem in hydrodynamic modeling