List of Figures

• 2.1. Shape of a shifted MAXWELL distribution function $f_\textrm {sM}(k)$ and its symmetric $f_\textrm {S}(k)$ and anti-symmetric $f_\textrm {A}(k)$ parts. The displacement is assumed to be large.
• 2.2. Shape of a shifted MAXWELL distribution function $f_\textrm {sM}(k)$ and its symmetric $f_\textrm {S}(k)$ and anti-symmetric $f_\textrm {A}(k)$ parts. The displacement is assumed to be small.
• 2.3. Symmetrical $f_\textrm {S}(k)$ and anti-symmetric $f_\textrm {A}(k)$ parts of a shifted MAXWELL distribution function in comparison with the result of the diffusion approximation.
• 2.4. Comparison of the different closure relations with the sixth moment from a Monte Carlo simulation.
• 2.5. Symbolic band diagram showing the four partial processes involved in indirect generation/recombination.
• 3.1. Schematic representation of an orthogonal mesh discretizing the active region of a MOSFET.
• 3.2. Three adjacent grid points together with some notational abbreviations used in the derivation.
• 3.3. A set of 13 grid points together with their associated VORONOI regions which are bounded by the dashed lines.
• 3.4. Control volume of grid point $P_i$ used for the box integration method.
• 3.5. Functional shape of the growth function $g(x)$ displayed in a normalized interval with $\alpha$ as parameter.
• 4.1. Sketch of the simulated SOI MOSFET including symbolic compact devices. Important effects are SHOCKLEY-READ-HALL generation/recombination (SRH) and impact-ionization (II).
• 4.2. Output characteristics of the SOI (Device 1) obtained by drift-diffusion simulations with and without impact-ionization.
• 4.3. Distributed potential of the SOI (Device 1) obtained by drift-diffusion simulations with impact-ionization turned on. The cutline through the device is located at a depth of $y = 100 \hspace {.35ex} \textrm {nm}$.
• 4.4. Distributed potential of the SOI (Device 1) obtained by drift-diffusion simulations with impact-ionization turned off. The cutline through the device is located at a depth of $y = 100 \hspace {.35ex} \textrm {nm}$.
• 4.5. Output characteristics of the SOI (Device 1) obtained by energy transport simulations using the device simulators MINIMOS-NT and DESSIS. The influence of impact-ionization is also shown.
• 4.6. Output characteristics of the SOI with a body contact (Device 2) obtained by energy transport simulations.
• 4.7. Bulk currents of the SOI with body contact (Device 2) obtained by energy transport simulations.
• 4.8. Comparison of the drain currents of the SOI (Device 1) and the device with body contact (Device 2) obtained by drift-diffusion simulations.
• 4.9. Distributed potential of the SOI (Device 1) obtained by energy transport simulations at a depth of $y = 100 \hspace {.35ex} \textrm {nm}$. The body potential drops below the equilibrium value of $- 0.46 \hspace {.35ex} \textrm {V}$.
• 4.10. Threshold voltage as a function of the body bias of the SOI with a body contact (Device 2) obtained by drift-diffusion simulations. The threshold voltage was defined as the gate-source voltage at which the drain current equals $0.1 \hspace {.35ex} \textrm {mA}$.
• 4.11. Electron concentration in the SOI (Device 1) obtained by a drift-diffusion simulation.
• 4.12. Electron concentration in the SOI (Device 1) obtained by an energy transport simulation.
• 4.13. SRH net-generation in the SOI (Device 1) obtained by a drift-diffusion simulation. Generation occurs only in the drain-body junction.
• 4.14. SRH net-generation in the SOI (Device 1) obtained by an energy transport simulation. Generation occurs in both junctions.
• 4.15. SRH net-recombination in the SOI (Device 1) obtained by a drift-diffusion simulation. Recombination occurs only in the source-body junction.
• 4.16. SRH net-recombination in the SOI (Device 1) obtained by an energy transport simulation. Recombination occurs in the whole p-body.
• 4.17. Body potential of the SOI (Device 1) obtained by a transient energy transport simulation.
• 4.18. Drain currents of the SOI (Device 1) obtained by a transient energy transport simulation showing different sweep times.
• 4.19. Body potentials of the SOI (Device 1) obtained by a transient energy transport simulation showing different sweep times.
• 4.20. Drain currents of the SOI (Device 1) obtained by a energy transport simulations showing different body dopings $\textrm {N}_\textrm {A}$.
• 5.1. Electron concentration in a MOSFET (Device 3) obtained by energy transport and Monte Carlo simulations.
• 5.2. Electron concentration in a MOSFET (Device 3) obtained by drift-diffusion and Monte Carlo simulations.
• 5.3. Comparison of the electron concentration in a MOSFET (Device 3) at a vertical cut located in the middle between source and drain obtained by drift-diffusion, energy transport, and Monte Carlo simulations.
• 5.4. Components of the temperature tensor compared to the temperature $T_n$ from the mean energy obtained by Monte Carlo simulations.
• 5.5. Temperature from the mean electron energy along the channel of a MOS transistor. Six characteristic points are marked for later reference.
• 5.6. Electron distribution function at six characteristics points along the channel of a MOS transistor. Note that the average energies for the points B and E are the same.
• 5.7. The distribution function at six characteristic points approximated by a MAXWELL distribution. Except for the contact regions the distribution function is never anything like a MAXWELLian.
• 5.8. The shape of the distribution function in four characteristic regions. (Picture gratefully taken from [69] with kind permission from the author.)
• 5.9. Distribution function in bulk for different electric field values.
• 6.1. Approximation of the anisotropic temperature by the analytical models.
• 6.2. Components of the temperature tensor obtained by Monte Carlo simulations compared to the analytical model of $T_{yy}$.
• 6.3. Kurtosis $\beta _n$ as a function of the temperature $T_n$ for bulk silicon with the doping concentration as a parameter together with the analytical expression for $\beta_{n, \mathrm{bulk}}$.
• 6.4. Monte Carlo simulation of an $n^+$-$n$-$n^+$ structure showing the normalized moment of fourth order $\beta _{n,\textrm {MC}}$ compared to the analytical $\beta _{n,1}$.
• 6.5. Monte Carlo simulation of an $n^+$-$n$-$n^+$ structure showing the hysteresis of the normalized moment of fourth order $\beta _{n,\textrm {MC}}$ compared to the analytical $\beta _{n,1}$.
• 6.6. Comparison of the non-MAXWELLian parameter obtained by Monte Carlo simulations and the empirical model $\beta _{n,2}$.
• 6.7. Shape of the functions used to model $\gamma _\nu$ and $\beta _n$. $\gamma _0$ and $\beta _0$ have been chosen to be $0.75$.
• 6.8. Monte Carlo simulation of a $90 \hspace {.35ex} \textrm {nm}$ and a $180 \hspace {.35ex} \textrm {nm}$ MOSFET (Device 3 with different gate-lengths) showing the $y$-component of the temperature tensor at the surface compared to the temperature $T_{n, \textsf {MC}}$ from the mean energy. The analytical model for $T_{yy}$ uses $\gamma _{0y} = 0.6$.
• 6.9. Monte Carlo simulation of a $90 \hspace {.35ex} \textrm {nm}$ and a $180 \hspace {.35ex} \textrm {nm}$ MOSFET (Device 3 with different gate-lengths) showing the normalized moment of fourth order $\beta _{n, \textsf {MC}}$ at the surface compared to the analytical model for $\beta _n$ with $\beta _0 = 0.75$.
• 6.10. Output characteristics of the SOI (Device 1) obtained by anisotropic energy transport simulations without closure modification ( $\beta _0 = 1$).
• 6.11. Output characteristics of the SOI (Device 1) assuming an anisotropic temperature ( $\gamma _{0y}=0.75$) and a modified closure relation at $V_\textrm {GS} = 1 \hspace {.35ex} V$.
• 6.12. Electron concentration in a MOSFET (Device 3) obtained by simulations using the modified energy transport model compared to Monte Carlo data.
• 6.13. Comparison of the electron concentration in a MOSFET (Device 3) at a vertical cut located in the middle between source and drain obtained by simulations using drift-diffusion (DD), standard energy transport (ET), Monte Carlo (MC), and the modified energy transport (MET) model.
• 6.14. Output characteristics of the "Well-Tempered" SOI (Device 4) at $V_\textrm {GS} = 1 \hspace {.35ex} V$.
• 6.15. Electron concentration in the "Well-Tempered" SOI (Device 4) obtained by a standard energy transport and a modified energy transport simulation.
• 6.16. Vertical potential distribution in the "Well-Tempered" SOI (Device 4) obtained by drift-diffusion, energy transport, and modified energy transport simulations.
• 6.17. Output characteristics of an SOI similar to Device 1 but with coarser grid in vertical direction.

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF