3-D self-consistent Schrödinger-Poisson solver: the spectral element method

In this paper, we developed an efficient three-dimensional (3-D) nanoelectronic device simulator based on a self-consistent Schrödinger-Poisson solver to simulate quantum transport. An efficient and fast algorithm, the spectral element method (SEM), is developed in this simulator to achieve spectral accuracy where the error decreases exponentially with the increase in the sampling density and the order of the polynomial basis functions, thus significantly reducing the CPU time and memory usage. Perfectly matched layer (PML) boundary method, as an alternative to the open-boundary conditions in NEGF, is applied in this solver to simplify the numerical implementation. The validity of the Schrödinger and Poisson solvers are illustrated by a multiple-terminal device and a spherical charge example, respectively. The utility of the self-consistent Schrödinger-Poisson solver is illustrated by a nanotube example.     

Quantum Monte Carlo Device Simulation of Nano-Scaled SOI-MOSFETs

Quantum transport properties of nano-scaled SOI-MOSFETs are investigated based on a quantum Monte Carlo (MC) device simulation. The quantum mechanical effects are incorporated in terms of a quantum correction of potential in the well-developed particle MC computational techniques. The ellipsoidal multi-valleys of silicon conduction band are also considered in the simulation. First, the validity of the quantum MC technique is verified by comparing the simulated results with a self-consistent Schrödinger-Poisson solution at thermal equilibrium. Then, we apply the technique to non-equilibrium and quasi-ballistic quantum transport in nano-scaled SOI-MOSFETs.     

VSP—a quantum-electronic simulation framework

The Vienna Schrödinger-Poisson (VSP) simulation framework for quantum-electronic engineering applications is presented. It is an extensive software tool that includes models for band structure calculation, self-consistent carrier concentrations including strain, mobility, and transport in transistors and heterostructure devices. The basic physical models are described. Through flexible combination of basic models sophisticated simulation setups for particular problems are feasible. The numerical tools, methods and libraries are presented. A layered software design allows VSP’s existing components such as models and solvers to be combined in a multitude of ways, and new components to be added easily. The design principles of the software are explained. Software abstraction is divided into the data, modeling and algebraic level resulting in a flexible physical modeling tool. The simulator’s capabilities are demonstrated with real-world simulation examples of tri-gate and nanoscale planar transistors, quantum dots, resonant tunneling diodes, and quantum cascade detectors.     

A fast and stable Poisson-Schrödinger solver for the analysis of carbon nanotube transistors

When the coupled Schrödinger-Poisson system is solved iteratively with appropriate numerical damping, convergence problems are likely to occur. We show that these problems are due to inappropriate energy discretization for evaluating the carrier concentration. By using an adaptive method the self-consistent loop becomes stable, and most of the simulations converge in a few iterations. We applied this approach to investigate the behavior of carbon nanotube field effect transistors.     

General 2D Schrödinger-Poisson solver with open boundary conditions for nano-scale CMOS transistors

Employing the quantum transmitting boundary (QTB) method, we have developed a two-dimensional Schrödinger-Poisson solver in order to investigate quantum transport in nano-scale CMOS transistors subjected to open boundary conditions. In this paper we briefly describe the building blocks of the solver that was originally written to model silicon devices. Next, we explain how to extend the code to semiconducting materials such as germanium, having conduction bands with energy ellipsoids that are neither parallel nor perpendicular to the channel interfaces or even to each other. The latter introduces mixed derivatives in the 2D effective mass equation, thereby heavily complicating the implementation of open boundary conditions. We present a generalized quantum transmitting boundary method that mainly leans on the completeness of the eigenstates of the effective mass equation. Finally, we propose a new algorithm to calculate the chemical potentials of the source and drain reservoirs, taking into account their mutual interaction at high drain voltages. As an illustration, we present the potential and carrier density profiles obtained for a (111) Ge NMOS transistor as well as the ballistic current characteristics.     

Accurate modeling of Metal/HfO<Subscript>2</Subscript>/Si capacitors

In this work, we study the differences caused in the Capacitance-Voltage (C-V) characteristics of MOS devices when SiO₂ is replaced by HfO₂ as the gate dielectric. A self-consistent Schrödinger-Poisson solver has been developed to include the effects of quantum confinement and the influence of different parameters such as the effective mass, barrier height, and dielectric constant (κ) of the gate insulator material. Two different devices are considered: A Double Gate MOSFET and a Surrounding Gate Transistor. The validity of the Equivalent Oxide Thickness (EOT) is studied.     

Microscopic simulation of RF noise in junctionless nanowire transistors

A deterministic solver for the analysis of microscopic noise and small-signal fluctuations in junctionless nanowire field-effect transistors is presented, which is based on a self-consistent and simultaneous solution of the Poisson/Schrödinger/Boltzmann equations. It is verified that the numerical framework fulfills the vital properties of reciprocity and passivity in the small-signal sense, and yields Johnson–Nyquist noise under equilibrium conditions. Key figures such as the cutoff frequency, drain excess noise factor, the Fano factor, and gate/drain correlation coefficient are presented at various bias conditions. In this work we show that similar to the inversion-mode MOSFETs, the gate and drain current noises mainly stem from the warm electrons at the source side, whereas the hot electrons do not have a significant contribution. Also, our results show that the device behaves similar to long-channel FETs in terms of its excess noise even for a channel length of 10 nm, due to the strong control of its electrostatics by the all-around gate.     

Simulation of resonant tunneling heterostructures: numerical comparison of a complete Schrödinger-Poisson system and a reduced nonlinear model

Two different models are compared for the simulation of the transverse electronic transport through a heterostructure: a 1D self-consistent Schrödinger-Poisson model with a numerically heavy treatment of resonant states and a reduced model derived from an accurate asymptotic nonlinear analysis. After checking the agreement at the qualitative and quantitative level on quite well understood bifurcation diagrams, the reduced model is used to tune double well configurations for which nonlinearly interacting resonant states actually occur in the complete self-consistent model.     

Quantum-mechanical effects in multiple-gate MOSFETs

In this work, a self-consistent solution of the 2D Schrödinger-Poisson equations is used to analyze Multiple-Gate MOSFETs. Classical simulations overestimate the peak density compared to quantum simulations and therefore the total electron density considered to calculate the current. The impact of the corner rounding on the electron distribution has also been analyzed. New devices, such as the Omega-gate MOSFETs have been studied as a function of the buried gate length.     

Semiclassical transport in silicon nanowire FETs including surface roughness

In this paper we investigate the effect of surface roughness scattering on transport in silicon nanowire FETs using a deterministic Boltzmann equation solver previously developed by the authors. We first solve the coupled Schrödinger-Poisson equations to extract the subband profiles along the channel, and then address the transport problem. Some features of the low-field mobility as a function of the wire diameter and gate bias are discussed and the effect of surface roughness on the I–V characteristics is presented.     

A Quantum Correction Model for Nanoscale Double-Gate MOS Devices Under Inversion Conditions

A quantum correction model for nanoscale double-gate MOSFETs under inversion conditions is proposed. Based on the solution of Schrödinger-Poisson equations, the developed quantum correction model is optimized with respect to (i) the left and right positions of the charge concentration peak, (ii) the maximum of the charge concentration, (iii) the total inversion charge sheet density, and (iv) the average inversion charge depth, respectively. This model can predict inversion layer electron density for various oxide thicknesses, silicon film thicknesses, and applied voltages. Compared to the Schrödinger-Poisson results, our model prediction is within 3.0% of accuracy. This quantum correction model has continuous derivatives and is therefore amenable to a device simulator.     

A multi-purpose Schrödinger-Poisson Solver for TCAD applications

We present the Vienna Schrödinger-Poisson Solver (VSP), a multi-purpose quantum mechanical solver for investigations on nano-scaled device structures. VSP includes a quantum mechanical solver for closed as well as open boundary problems on fairly arbitrary one-dimensional cross sections within the effective mass framework. For investigations on novel gate dielectrics VSP holds models for bulk and interface trap charges, and direct and trap assisted tunneling. Hetero-structured semiconductor devices, like resonant tunneling diodes (RTD), can be treated within the closed boundary model for quick estimation of resonant energy levels. The open boundary model allows evaluation of current voltage characteristics.     

The impact of high-k gate dielectric and FIBL on performance of nano DG-MOSFETs with underlapped source/drain regions

The impact of high-k gate dielectrics and fringing induced barrier lowering (FIBL) effects on a nano double gate MOSFET is studied over a wide range of dielectric permittivity using ballistic quantum simulation. The simulations are based on self-consistent solution of 2D Poisson equation and Schrödinger equation with open boundary conditions, within the Non-equilibrium Green’s Function formalism. The numerical results show that the use of high-k gate at fixed equivalent oxide thickness (EOT), deteriorates the short channel effects due to FIBL effect. We show that the FIBL can be effectively suppressed by using underlapped source/drain region.     

Nanowire transistor modeling: influence of ionized impurity and correlation effects

This study presents two relevant effects influencing the electronic transport of nanowire transistors. We first focus on the ionized impurity impacts and calculate the current characteristics with a self-consistent three-dimensional (3D) Green’s function approach. The results show the effects of both acceptor and donor impurities on the physical electron properties. In particular, we emphasize that the presence of a donor induces different transport phenomena according to the applied gate bias. In a second part, we report a numerical study of the self-energy correction due to correlation effects from dynamic screening of the moving electron in silicon nanowire transistors. This many-body effect, which is not included in the usual Hartree approximation, is then incorporated self-consistently into a non-equilibrium Green’s function (NEGF) code. The results pinpoint the importance of dielectric confinement whose magnitude can not be neglected compared to its quantum counterpart in ultimate nanowire transistors.     

Scalability and process induced variation analysis of polarity controlled silicon nanowire transistor

In this paper, we present scalability and process induced variation analysis of polarity gate silicon nanowire field-effect transistor. 3D simulation results show that the PGFET offers significant reduction in short channel effects and variability due to utilization of uniform lightly doped silicon nanowire (SiNW) as compare to highly doped silicon nanowire in junctionless transistors. The performance parameters were evaluated for different device geometries, such as variation in SiNW radius, equivalent oxide thickness, channel length and spacer length. Sensitivity analysis shows that the PGFET exhibits less dependence towards gate length in comparison to other device parameters. It is seen that ON to OFF current ratio variation with silicon nanowire thickness is lower for PGFET as compared to JLFET. The threshold voltage roll-off and sensitivity towards intrinsic delay in PGFET is much lower than its counterpart device.     

A study of threshold voltage fluctuations of nanoscale double gate metal-oxide-semiconductor field effect transistors using quantum correction simulation

In this paper, we computationally investigate fluctuations of the threshold voltage introduced by random dopants in nanoscale double gate metal-oxide-semiconductor field effect transistors (DG MOSFETs). To calculate variance of the threshold voltage of nanoscale DG MOSFETs, a quantum correction model is numerically solved with the perturbation and the monotone iterative techniques. Fluctuations of the threshold voltage resulting from the random dopant, the gate oxide thickness, the channel film thickness, the gate channel length, and the device width are calculated. Quantum mechanical and classical results have similar prediction on fluctuations of the threshold voltage with respect to different designing parameters including dimension of device geometry as well as the channel doping. Fluctuation increases when the channel doping, the channel film thickness, and/or the gate oxide thickness increase. On the other hand, it decreases when the channel length and/or the device width increase. Calculations of the quantum correction model are quantitatively higher than that of the classical estimation according to different quantum confinement effects in nanoscale DG MOSFETs. Due to good channel controllability, DG MOSFETs possess relatively lower fluctuation, compared with the fluctuation of single gate MOSFETs (less than a half of the fluctuation[-11pc] of SG MOSFETs). To reduce fluctuations of the threshold voltage, epitaxial layers on both sides of channel with different epitaxial doping are introduced. For a certain thickness of epitaxial layers, the fluctuation of the threshold voltage decreases when epitaxial doping decreases. In contrast to conventional quantum Monte Carlo approach and small signal analysis of the Schrödinger-Poisson equations, this computationally efficient approach shows acceptable accuracy and is ready for industrial technology computer-aided design application.     

Monte Carlo Simulations of THz Quantum-Cascade Lasers

This work is a theoretical study, compared with experimental data, on GaAs/Al_0.15Ga_0.85 As quantum-cascade laser structure operating at λ = 66 μm. The Monte Carlo method is used to give useful information about the electron dynamics in such structures, by extracting the electron population and distribution function at every step of the simulation. Electron wavefunctions and energies are calculated using a self-consistent non-parabolic Schrödinger-Poisson solver. Longitudinal Optical phonon, acoustic phonon and electron-electron scatterings are included. Finally non-equilibrium phonons have been explicitely considered and the Pauli's exclusion principle is taken into account. In addition, electroluminescence spectra and transition rates are calculated. Results show that population inversion occurs for an electric field F = 3.15 KV/cm at a temperature of T = 80 K. The electrons involved in the lasing action come from two different levels very close in energy. The electronic temperatures extracted from the distribution functions agree very well to the measured ones. The rates extracted during the simulation allow us to give a complete insight on the mechanisms responsible for the population and depletion of the laser levels.