List of Figures

• 1.1. Hierarchy of transport models
• 1.2. Quantum confinement in a MOSFET structure
• 1.3. For increasing subbands the difference between the energy eigenvalues decreases and converge to zero for an infinite number of energy-eigenstates. In this limit the subband system becomes a bulk system.
• 1.4. Conduction band and the first two wavefunctions of a thin film SOI MOSFET for different gate voltages. For increasing gate voltages the wavefunctions are shifted towards the interface.
• 1.5. Fermi-Dirac distribution function for $\mathrm {0 K}$ , $\mathrm {300 K}$ , $\mathrm {500 K}$ , and the limit $k_\ensuremath {\mathrm {B}}\ensuremath {T}\ll \ensuremath {\mathcal {E}}-\ensuremath {\mathcal {E}}_\mathrm {f}$ is demonstrated. In this limit, the Fermi-Dirac function can be approximated by a Maxwell distribution function.
• 1.6. Scattering event from one trajectory to another in phase space. Scattering events, which are assumed to happen instantly, change the carrier's momentum, while the position is not affected (after [1]).
• 1.7. Energy ellipsoids of the first conduction band within the first Brillouin zone of silicon (after [2]).
• 1.8. Unprimed and primed subband ladders
• 1.9. Populations of the first two subbands in the unprimed, primed, and double primed valleys versus lateral field in a UTB SOI MOSFET test device. Relative occupations are shifted to higher subbands in each valley for higher fields.
• 1.10. Velocities of the first and second subband of the unprimed, primed, and double primed valleys as well as the average total velocity versus lateral electric field.
• 1.11. The electron concentration of a single gate SOI MOSFET has been calculated classically, quantum-mechanically, together with the quantum correction models MLDA, Van Dort, and the improved MLDA (after [3]).
• 1.12. Kurtosis for a $\mathrm {100 nm}$ $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure calculated with the MC method. In the channel the kurtosis is lower than one, which means that the heated Maxwellian overestimates the carrier distribution function, while the Maxwellian underestimates the carrier distribution in the drain.
• 1.13. Ratio between the sixth moment obtained from three-dimensional bulk MC simulation and the analytical closure relation (1.124) of the six moments model for different values of  $\ensuremath {c}$ (see left part). The maximum peak at point B of the ratio as a function of the lattice temperature is shown on the right.
• 1.14. Distribution function at point B for lattice temperatures of $\mathrm {200 K}$ , $\mathrm {300 K}$ , and $\mathrm {400 K}$ . The high energy tail of the carrier distribution function decreases for high lattice temperatures.
• 1.15. The ratio of the six moments model obtained from two-dimensional Subband Monte Carlo data with the analytical 2D closure relation of the six moments model for different  $\ensuremath {c}$ is presented. As can be observed is for the 2D case as well the best value.
• 1.16. $H_{1}\mathrm {,}$ $H_{2}\mathrm {,}$ and $H_{3}$ as functions of the energy with an effective field of 950 kV/cm. For low energies, the non-parabolicity factors approach unity. The non-parabolicity factors have been calculated out of Subband Monte Carlo simulations.
• 1.17. Flowchart of a MC Simulation (after [1])
• 1.18. $Y_{10}$ and $Y_{11}$ in polar coordinates
• 1.19. A comparison of the low and high-field mobility as a function of the doping in the bulk calculated with the SHE method and the drift-diffusion model. In the SHE simulation, only the first Legendre polynomial has been taken into account.
• 1.20. The velocity profile of an $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure calculated with a device MC simulation and with a SHE simulator taking 1, 5, 9, and 15  Legendre polynomials (LP) into account.
• 2.1. Carrier mobility $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ , energy flux mobility $\ensuremath {\ensuremath {\mu }_{\mathrm {1}}}$ , and second-order energy flux mobility $\ensuremath {\ensuremath {\mu }_{\mathrm {2}}}$ versus driving field for different doping concentrations. For fields higher than ${\mathrm{100}}{\;}{\mathrm{kV/cm}}$ , the mobilities are independent of the doping concentration, while for low fields the values of the mobilities of the low doping case is high compared to high doping concentrations.
• 2.2. Energy-relaxation time  $\ensuremath {\tau _\mathrm {1}}$ and second-order energy relaxation time  $\ensuremath {\tau _\mathrm {2}}$ extracted from bulk MC simulations as a function of the kinetic energy for different bulk dopings. For very high energies, the relaxation times decrease due to the increase of optical phonon scattering.
• 2.3. Bulk velocity of electrons as a function of the driving field  $\ensuremath {E}_{\mathrm {abs}}$ for a doping of ${\mathrm{10^{14}}}{\;}{\mathrm{cm^{-3}}}$ , ${\mathrm{10^{16}}}{\;}{\mathrm{cm^{-3}}}$ , and ${\mathrm{10^{18}}}{\;}{\mathrm{cm^{-3}}}$ . In the low field regime, the electron velocity for high dopings is lower than the velocity of the low dopings, while the value of the velocity converges for high fields.
• 2.4. Output currents for different  $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structures calculated with DD, ET, and SM models. As a reference, SHE simulations are used. For ${\mathrm{1000}}{\;}{\mathrm{nm}}$ , all models predict the same current, while the DD model underestimates the current for a channel length of ${\mathrm{100}}{\;}{\mathrm{nm}}.$
• 2.5. Relative error of the current calculated with the DD, ET, and the SM model as a function of the channel length. A voltage of $\mathrm {1 V}$ has been applied. While the ET and SM model is below $\mathrm {7.5 \%,}$ the DD model approaches to $\mathrm {16 \%}$ at a channel length of $\mathrm {100 nm}$ .
• 2.6. Velocity profiles of a  $\mathrm {50 nm}$ , $\mathrm {100 nm}$ , and $\mathrm{200 nm}$ long  $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure calculated with the MC method are presented after [4]. The velocity overshoot at the beginning of the lowly doped n-region is clearly visible.
• 2.7. Carrier temperature  $\ensuremath {T_\ensuremath {n}}$ as a function of the driving field in a homogeneous bulk simulation carried out with fullband MC. For lower fields, the carrier temperature is a function of  $\ensuremath {E}^2$ , while for high fields, the temperature is a linear function of the driving field.
• 2.8. Evolution of the distribution function inside an $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure. The mixture of hot and cold electrons is expressed by the high-energy tail of the carrier distribution function.
• 2.9. Kurtosis calculated for different source and drain doping concentrations in an $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure with a channel length of ${\mathrm{100}}{\;}{\mathrm{nm}}$ . The maximum peak of the kurtosis for high doping concentrations is very high compared to the low doping case.
• 2.10. Carrier temperature  $\ensuremath {T_\ensuremath {n}}$ together with second-order temperature  $\ensuremath {\Theta }$ for a  ${\mathrm{1000}}{\;}{\mathrm{nm}}$ and a  ${\mathrm{60}}{\;}{\mathrm{nm}}$ device. A field of  ${\mathrm{50}}{\;}{\mathrm{kV/cm}}$ has been assumed. While in the long channel device a Maxwellian can be used, the high-energy tail in the short channel device in the drain region increases.
• 2.11. Distribution function at point D of Fig. 2.8 for $\mathrm {40 nm}$ , $\mathrm {60 nm}$ , $\mathrm {80 nm}$ , and $\mathrm {100 nm}$ channel devices. As can be observed, the high-energy tail for increasing channel lengths decrease.
• 2.12. Kurtosis  $\ensuremath {\beta }$ and the carrier temperature for electric fields of ${\mathrm{5}}{\;}{\mathrm{kV/cm}}$ , ${\mathrm{20}}{\;}{\mathrm{kV/cm}}$ , and ${\mathrm{50}}{\;}{\mathrm{kV/cm}}$ through a  ${\mathrm{100}}{\;}{\mathrm{nm}}$ channel $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ device (the value of the fields in the upper left part is calculated at point A). For high fields, the kurtosis increases at the beginning of the drain region, which means that the high-energy tail of the distribution function is becoming very important. In the right upper part, the carrier temperature profile for different electric fields is shown. The kurtosis exceeds unity, while the carrier temperature drops down. The velocity profile for fields of ${\mathrm{5}}{\;}{\mathrm{kV/cm}}$ and ${\mathrm{50}}{\;}{\mathrm{kV/cm}}$ is shown on the lower part. For low fields, all models yield the same velocity profiles, which is an indication that the heated Maxwellian can be used. For high fields, a significant deviation of the velocity profiles can be observed.
• 2.13. The impact ionization rate is calculated with MC, the DD, ET, and the SM model for a  $\mathrm {200 nm}$ and a $\mathrm {50 nm}$ structure. Due to the better modeling of the distribution function in the SM model, the results are closer to the MC data than the DD and the ET model (after [5]).
• 2.14. Evolution of the carrier velocity profiles for decreasing channel lengths calculated with the DD, ET, and the SM model. The velocities are compared to the results obtained from SHE simulations. While the maximum velocity of the DD model is the saturation velocity  $\ensuremath {v_\mathrm {sat}}$ , the spurious velocity overshoot at the end of the channel in the ET and the SM model is clearly visible. The velocity overshoot at the beginning of the channel can be quantitatively identified at the  ${\mathrm{100}}{\;}{\mathrm{nm}}$ device in the ET and the SM model.
• 2.15. Velocity profile calculated with ET model and MC data. Due to the MC closure in the ET model for the fourth order moment and due to the improved modeling of the transport parameters, the spurious velocity overshoot at the end of the channel disappears (after [6]).
• 2.16. Output currents of a  $\mathrm {80 nm}$ and a  $\mathrm {40 nm}$ channel length  $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure calculated with the DD, ET, SM, and SHE model. The ET model overestimates the current at $\mathrm {40 nm}$ , while the SM model yields the most accurate result.
• 2.17. Relative error in the current of the DD, ET, and the SM model for an  $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure in the channel range from  ${\mathrm{100}}{\;}{\mathrm{nm}}$ down to  ${\mathrm{40}}{\;}{\mathrm{nm}}$ . While the relative error of the SM model is below  ${\mathrm{6}}{\;}{\mathrm{\%}}$ , the error of the DD and the ET model is at  ${\mathrm{-30}}{\;}{\mathrm{\%}}$ and  ${\mathrm{40}}{\;}{\mathrm{\%}}$ for a channel length of  ${\mathrm{40}}{\;}{\mathrm{nm}}$ , respectively.
• 3.1. Principle data flow of the parameter extraction for higher-order transport models. While transport is treated in the SMC code, the influence of the confinement perpendicular to the oxide interface is carried out by the Schrödinger-Poisson solver.
• 3.2. Carrier velocity as a function of the driving field for different inversion layer concentrations. Due to surface roughness scattering, the carrier velocity decreases for increasing inversion layer concentrations. The maximum velocity is below the saturation velocity of bulk Si.
• 3.3. The effective mobility as a function of the effective field [7]. For increasing bulk doping and for low inversion layer concentrations Coulomb scattering is the main scattering process, while for increasing  $\ensuremath {N}_\mathrm {inv}$ phonon scattering becomes more important.
• 3.4. Higher-order mobilities as a function of the inversion layer concentration  $\ensuremath {N}_\mathrm {inv}$ for different lateral fields. For high fields the difference of the mobilities decreases. For low fields in a bulk MOSFET the carrier mobility is equal to the measurement data of Takagi.
• 3.5. Conduction band edge as a function of the position for different inversion layer concentrations. For high  $\ensuremath {N}_\mathrm {inv}$ , the carriers are closer to the interface and hence they are more affected by surface roughness scattering than for low  $\ensuremath {N}_\mathrm {inv}$ .
• 3.6. Conduction band and wavefunctions of a UTB SOI MOSFET for different electric fields. Both the wavefunctions and subbands are shifted with increasing lateral electric fields. The conduction band edge is affected by the change in the subband occupations.
• 3.7. Influence of surface roughness scattering on $\ensuremath {\ensuremath {\mu }_{\mathrm {0}}}$ , $\ensuremath {\ensuremath {\mu }_{\mathrm {1}}}$ , and $\ensuremath {\ensuremath {\mu }_{\mathrm {2}}}$ as a function of the lateral field for different $\ensuremath {N}_\mathrm {inv}$ . For low fields, surface roughness scattering has a strong impact, while for high fields the mobilities are unaffected by SRS.
• 3.8. Ratio between the mobilities neglecting and considering SRS for high and low  $\ensuremath {N}_\mathrm {inv}$ values as a function of the lateral field. For low  $\ensuremath {N}_\mathrm {inv}$ , the mobilities are unaffected. The carrier mobility is more affected by SRS than the higher-order mobilities.
• 3.9. Ratio of the momentum relaxation time  $\ensuremath {\tau _\mathrm {0}}$ and energy flux relaxation time  $\ensuremath {\tau _\mathrm {3}}$ for different  $\ensuremath {N}_\mathrm {inv}$ as a function of the driving field. Surface roughness scattering influences  $\ensuremath {\tau _\mathrm {0}}$ more than  $\ensuremath {\tau _\mathrm {3}}$ , especially for high inversion layer concentrations.
• 3.10. Influence of surface roughness scattering on the electron velocities for different  $\ensuremath {N}_\mathrm {inv}$ . For decreasing  $\ensuremath {N}_\mathrm {inv}$ , SRS loses the influence on the carrier velocity. Due to non-parabolic bands and quantization, the velocity is below the saturation velocity of the bulk.
• 3.11. Influence of surface roughness scattering on  $\ensuremath {\tau _\mathrm {1}}$ and $\ensuremath {\tau _\mathrm {2}}$ as a function of the kinetic energy of the carriers for different  $\ensuremath {N}_\mathrm {inv}$ . Due to the elastic scattering nature of SRS, the relaxation times are not affected by SRS.
• 3.12. Energy relaxation time (left side) and the second-order relaxation time (right side) as a function of the inversion layer concentration for different lateral electric fields. For a field of $\mathrm {50 kV/cm}$ and high $\ensuremath {N}_\mathrm {inv}$ , $\ensuremath {\tau _\mathrm {1}}$ and $\ensuremath {\tau _\mathrm {2}}$ increase in contrast to high-fields, where the energy relaxation time remain constant.
• 3.13. First subband occupation of the unprimed, primed, and double primed valleys as a function of the inversion layer concentration for fields of  $\mathrm {50 kV/cm}$ and  $\mathrm {100 kV/cm}$ . Due to the light mass of the unprimed valley in transport direction, the subband occupation number is higher than in the other valleys.
• 3.14. Mobilities of the unprimed, primed, and double primed valleys, and total average mobility as a function of the channel thickness. At about $\mathrm {3 nm}$ , there is a maximum in the total mobility. This is the point where the mobility of the unprimed valley is already high while the carrier mobility of the remaining valleys is low.
• 3.15. Populations of the unprimed, primed, and double primed valleys as functions of the channel thickness.
• 3.16. Carrier, energy-flux, and second-order energy flux mobility of SMC and bulk MC simulations. The mobilities obtained by bulk simulations are higher than in subband simulations. For high fields, the mobilities from subband simulations yield the same value as from bulk simulations.
• 3.17. Comparison of the energy relaxation time and second-order energy relaxation time using 2D SMC data and 3D bulk MC data. For high energies both simulations converge to the same value.
• 3.18. Comparison of the carrier temperature using 2D SMC data and 3D bulk MC data. Due to higher phonon scattering in the 2D case the temperature is lower than in the 3D case.
• 3.19. Subband occupations as functions of the subband ladders in the unprimed, primed, and double primed valleys for electric fields of  $\mathrm {1 kV/cm}$ and  $\mathrm {100 kV/cm}$ . For high fields, the subband ladders in the primed and double primed valleys are occupied.
• 4.1. The SP-SMC loop describes the transport of a two-dimensional electron gas in an inversion layer. After convergency is reached, the device simulator utilizes the extracted parameters to characterize transport through the channel of the whole device.
• 4.2. Effective field profile throughout the whole device for several bias points. With the effective fields and the SMC tables, higher-order transport parameters can be modeled as a function of the effective field.
• 4.3. Energy relaxation time and second-order relaxation time for different effective fields as a function of the kinetic energy of the carriers. For high carrier energies, the relaxation times of the different inversion layers yield the same value.
• 4.4. Carrier and higher-order mobilities for different effective fields as a function of the lateral field. For high fields the mobilities converge to the same value.
• 4.5. Second-order temperature $\theta =\ensuremath {\beta }\ensuremath {T_\ensuremath {n}}$ , carrier temperature $\ensuremath {T_\ensuremath {n}}$ , and kurtosis  $\ensuremath {\beta }$ for different SOI MOSFETs with channel lengths of ${\mathrm{100}}{\;}{\mathrm{nm}}$ , ${\mathrm{60}}{\;}{\mathrm{nm}}$ , and ${\mathrm{40}}{\;}{\mathrm{nm}}$ . For decreasing channel lengths the kurtosis increases due to the increase of the high energy tail of the distribution function.
• 4.6. Temperature and second-order temperature profiles for different drain voltages. For the low drain voltage case, the second-order temperature yields a similar result compared to the carrier temperature, while a significant deviation between $\theta$ and $\ensuremath {T_\ensuremath {n}}$ especially in the drain region can be observed for high fields.
• 4.7. Capacity versus gate voltages for devices with  $\mathrm {40 nm}$ , $\mathrm {60 nm}$ , and  $\mathrm {80 nm}$ gate lengths calculated with the Schrödinger-Poisson solver and with the calibrated quantum correction model. For a gate voltage used in most simulations of $\mathrm {1.3 V}$ both simulators yield the same result.
• 4.8. Output characteristics of a  $\mathrm {40 nm}$ channel length UTB SOI MOSFET calculated with the DD, ET, SM models and, as a reference, with DSMC data. As can be observed, the SM model delivers a current very close to the SMC current. Neglecting the quantum correction model increases the output current of the macroscopic models.
• 4.9. Output current of $\mathrm {1000 nm}$ and $\mathrm {100 nm}$ channel devices calculated with the DD, ET, and SM model are compared to the output current obtained by DSMC simulations. For the $\mathrm {1000 nm}$ device, the results of all models converge.
• 4.10. Output current of a $\mathrm {30 nm}$ channel length device calculated with the DD, ET, SM models, and with the SMC method. The SM model predicts the most accurate result, while ET overestimates and DD underestimates the current, respectively.
• 4.11. Evolution of the velocity profiles of a UTB SOI MOSFET with a channel length of $\mathrm {30 nm}$ for drain voltages of $\mathrm {V}_{\mathrm {d}}=\mathrm {0.2 V}$ , $\mathrm {V}_{\mathrm {d}}=\mathrm {0.4 V}$ , $\mathrm {V}_{\mathrm {d}}=\mathrm {0.6 V}$ , and $\mathrm {V}_{\mathrm {d}}=\mathrm {0.8 V}$ . The spurious velocity overshoot, especially in the ET model is clearly visible for drain voltages of  $\mathrm {0.8 V}$ . The SM model predicts most accurate results.
• 4.12. Output current at $V_d = \mathrm {1 V}$ as a function of the channel length. A significant increase in the current of the ET model at channel lengths below $\mathrm {80 nm}$ can be observed, while the current from the DD model is below the current of the DSMC. The SM model yields the most accurate current.
• 4.13. Relative error as a function of the channel length of the DD, ET, and the SM models. The error of the ET model increases rapidly for devices with a channel length below $\mathrm {80 nm}$ where even the DD model becomes better. The SM model is the most accurate model for short channel devices.
• 4.14. Transit frequencies as a function of the channel length. A significant increase of the frequency in the ET model at channel lengths below $\mathrm {80 nm}$ can be observed, while the frequency from the DD model is below the frequency of the DSMC. The SM model yields the most accurate result.
• 4.15. Relative error of the transit frequencies as a function of the channel length of the DD, ET, and the SM models. The error of the DD model is higher than the error of the current (see Fig. 4.13), while the SM model is here as well the most accurate model.
• 4.16. Carrier and higher-order mobilities for a $\mathrm {40 nm}$ channel length device. The influence of SRS at the beginning of the channel is stronger than at the end.
• 4.17. Carrier temperature $\ensuremath {T_\ensuremath {n}}$ and second-order temperature  $\ensuremath {\Theta }$ calculated once with MC tables considering SRS and neglecting SRS, respectively. As can be seen $\ensuremath {T_\ensuremath {n}}$ and $\ensuremath {\Theta }$ are unaffected by SRS.
• 4.18. Output characteristics of a $\mathrm {40 nm}$ channel length SOI MOSFET calculated with the DD, ET, and the SM model using SMC data with SRS and SMC data without SRS.
• 4.19. Velocity profile of a $\mathrm {40 nm}$ channel length SOI MOSFET computed with the two-dimensional DD, ET, and SM model neglecting SRS in the subband MC tables and with the 3D macroscopic models using fullband MC tables.
• 4.20. Output characteristics of a $\mathrm {40 nm}$ channel length SOI MOSFET calculated once with the 2D macroscopic models using SMC data without SRS and with their 3D counterpart using fullband MC data.
• 5.1. Carrier mobility and higher-order mobilities for a $\mathrm {10 \%}$ and $\mathrm {40 \%}$ Ge composition, respectively as a function of the lateral electric field. Ge has got a deep influence on the mobilities for low fields, while for high-fields the influence of Ge on the mobilities decreases.
• 5.2. Low field carrier mobility together with higher-order mobilities as a function of the Ge composition. The carrier mobility fits the data from [8] quite well. The minimum of the mobilities is at $\mathrm {40 \%}$ , while a high increase of the mobility values can be observed for Ge composition greater than $\mathrm {80 \%}$ .
• 5.3. The energy relaxation time (left) and the second-order relaxation time (right) for a Ge mole fraction of $\mathrm {10 \%}$ and $\mathrm {40 \%}$ as a function of the carrier energy. For the $\mathrm {10 \%}$ Ge case the relaxation times are lower than for the $\mathrm {40 \%}$ Ge case.
• 5.4. Carrier velocities for different doping concentrations as a function of the electric field and for different Ge compositions. For high fields the velocities are independent of the doping concentrations, while the velocities for low Ge compositions are higher than for the $\mathrm {40 \%}$ case.
• 5.5. Kurtosis for a $\mathrm {100 nm}$ $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure for Si and SiGe. In the channel region the kurtosis of SiGe is higher than in Si, while the kurtosis of Si exceeds the one of SiGe at the beginning of the drain region.
• 5.6. Sixth moment obtained by MC simulations divided by the expression from equation (1.124) for different values of  $\ensuremath {c}$ . The value 2.7 is also a very good choice in SiGe and is even improved compared to Si.
• 5.7. Carrier mobility and higher-order mobilities for GaAs as a function of the lateral electric field. The values of the mobilities are very high with respect to the ones for Si.
• 5.8. Energy relaxation time and the second-order relaxation time for GaAs obtained for different doping concentrations as a function of the kinetic energy. The energy relaxation time has its maximum at $\mathrm {100 meV}$ , while the shape of the second-order energy relaxation time is completely different compared to the energy relaxation time.
• 5.9. The carrier velocity for different doping concentrations as a function of the driving field. For a field of $\mathrm {5 kV/cm}$ the carrier velocity has its maximum, while for high fields the velocity decreases.
• 5.10. $\Gamma$ - and L-valleys of GaAs are shown. For low fields the carriers are in the light hole $\Gamma$ -valley, while for high-fields the carriers are in the upper heavy hole valleys, decreasing the carrier velocity (after [9]).
• 5.11. Kurtosis profile of the $\mathrm {n}^+\mathrm {n}\mathrm {n}^+$ structure with a channel length of ${\mathrm{100}}{\;}{\mathrm{nm}}$ of GaAs and Si. The kurtosis at the end of the channel is lower in GaAs than in Si, while the kurtosis of GaAs is beyond Si at the beginning of the drain region.
• 5.12. Sixth moment obtained by MC simulations divided by the analytical expression from equation (1.124) for different values of  $\ensuremath {c}$ . The lower-order moments for equation (1.124) have been taken from MC simulations. Note that $\ensuremath {c}=\mathrm {2.7}$ is also a good choice in GaAs.