Self-consistent quantum transport theory: Applications and assessment of approximate models

We have implemented a fully self-consistent non-equilibrium Green’s function approach for vertical quantum transport in open quantum devices with contacts and study theoretically quantum well heterostructures, resonant tunneling diodes and quantum cascade laser structures in this formalism. We systematically investigate the role and consequences of several widely used approximations such as decoupling the equations for the scattering states and their occupation, neglect of inelastic scattering, and neglect of nonlocal scattering self-energies.     

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.     

Parallelization of the Nanoelectronic Modeling Tool (NEMO 1-D) on a Beowulf Cluster

NEMO's main task is the computation of current-voltage (I-V) characteristics for resonant tunneling diodes (RTDs). The primary model for high performance RTDs is the full band sp3s* tight binding simulation, which is based on a numerical double integral of energy and transverse momentum over a transport kernel at each bias point. A full charge self-consistent simulation invoking this model on a single CPU is prohibitively expensive, as the generation of a single I-V curve would take about 1–2 weeks to compute. Simplified charge self-consistent models, eliminating the numerical momentum integral for the quantum mechanical charge self-consistency, followed by a single pass double integration for the current, have been used in the past. However, Computation on a parallel computer now enables the thorough exploration of quantum mechanical transport including charge self-consistency effects within the entire Brillouin zone based on the double integral. Various parallelization schemes (fine, coarse, and mixed) are presented and evaluated in their performance. Finally a comparison to experimental data is given.     

NEGF simulation of the RTD bistability

The simulation of I-V characteristics of Al_0.3Ga_0.7As-GaAs and AlAs-GaAs resonant tunneling diodes (RTD) is presented. The nonequilibrium Green function (NEGF) based 1D quantum transport simulator Wingreen is used in our case. The plateau region on the IV characteristics usually present only by the Wigner function equation (WFE) based simulation appeared now by the NEGF simulation of our AlAs-GaAs RTD and its shape is comparable with our experimental measurements. Analysis of our results from point of view of the scattering and geometrical parameters of the RTD structure is presented.     

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.     

Shot noise in resonant tunneling diodes using the non-equilibrium Green’s functions calculation

The non-equilibrium Green function (NEGF) technique is used to solve the quantum transport equation in resonant tunneling diodes (RTDs). The charge interaction is treated self-consistently to include rigorously the space-charges effects. Reasonable results for the potential profile, the transmission probability, and the current-voltage characteristics have been obtained. The effect of temperature on the current-voltage (I-V) characteristics is investigated. Particularly, the current noise spectral density has been extracted following both coherent and sequential tunneling approaches. Our conclusion of the dominance of the coherent and sequential tunneling according to the transport regime is consistent with recent theoretical analyses and experimental data.     

The Use of Quantum Potentials for Confinement and Tunnelling in Semiconductor Devices

As MOSFETs are scaled to sub 100 nm dimensions, quantum mechanical confinement in the direction normal to the silicon dioxide interface and tunnelling (through the gate oxide, band-to-band and from source-to-drain) start to strongly affect their characteristics. Recently it has been demonstrated that first order quantum corrections can be successfully introduced in self-consistent drift diffusion-type models using Quantum Potentials. In this paper we describe the introduction of such quantum corrections within a full 3D drift diffusion simulation framework. We compare the two most popular quantum potential techniques: density gradient and the effective potential approaches, in terms of their justification, accuracy and computational efficiency. The usefulness of their 3D implementation is demonstrated with examples of statistical simulations of intrinsic fluctuation effects in decanano MOSFETs introduced by discrete random dopants. We also discuss the capability of the density gradient formalism to handle direct source-to-drain tunnelling in sub 10 nm double-gate MOSFETS, illustrated in comparison with Non-Equilibrium Green's Functions simulations.     

Proximity Effect of the Contacts on Electron Transport in Mesoscopic Devices

The Wigner function formalism is used for studying electron quantum transport in mesoscopic systems. In this work we show that, if the correlation of the electron wave function vanishes outside the region of interest (for example inside the contacts), then transport is affected inside the device. This property is verified analytically. Results show that, for very short devices, tunneling is actually influenced by the distance between the contacts. Modification in the electron density and conductivity have been numerically observed.     

Quantum Monte Carlo simulation of dissipative transport using Bohmian trajectories

In this paper, a Monte Carlo method is proposed, which utilizes Bohmian trajectories to simulate dissipative transport in one-dimensional quantum devices. The proposed method, similar to the classical Monte Carlo method, is capable of simulating both elastic and inelastic scattering effects, with the distinction that quantum effects such as tunneling are also included. At first, the Bohmian trajectories for the wave packets injected from the right and the left contacts are obtained by solving the time-dependent Schrodinger equation, and then scattering effects are included via stochastic changes applied on the electron trajectories. We have shown that the results of the proposed model agree well with those of NEGF formalism.     

Quantum Aspects of Resolving Discrete Charges in ‘Atomistic’ Device Simulations

In ‘atomistic’ device simulation the resolving of discrete charges onto a fine grained simulation mesh can lead to problems. The sharply resolved coulomb potential can cause simulation artefacts to appear in classical simulation environments using Boltzmann or Fermi-Dirac statistics. Various methods have been proposed in an effort to reduce or eliminate artefacts such as the trapping of mobile carriers in sharply resolved Coulomb wells, however they have met with limited success. In this paper we show an alternative approach for handling discrete charges in drift diffusion ‘atomistic’ simulations by properly introducing the related quantum mechanical effects using the Density Gradient formalism. This produces the desired effect of eliminating the trapping of mobile charge in heavily doped regions of the device.     

Smooth Quantum Hydrodynamic Model vs. NEMO Simulation of Resonant Tunneling Diodes

The smooth quantum hydrodynamic model is an extension of the classical hydrodynamic model for semiconductor devices which can handle in a mathematically rigorous way the discontinuities in the classical potential energy which occur at heterojunction barriers in quantum semiconductor devices. Smooth QHD model simulations of the current-voltage curves of resonant tunneling diodes are presented which exhibit negative differential resistance—the experimental signal for quantum resonance effects—and are compared with the experimentally verified current-voltage curves predicted by the simulator NEMO, which uses a non-equilibrium Green function method.The NEMO work described in this article was carried out in part at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration. This material is also based upon work supported by the National Science Foundation under Grant No. EEC-0228390.     

RTD Relaxation Oscillations, the Time Dependent Wigner Equation and Phase Noise

Wigner simulations of resonant tunneling diode (RTD) self-excited oscillations are discussed with respect to the upper frequency limit of operation and their sensitivity to large scale perturbations. These studies offer the most practical assessment of phase noise, response times of RTDs and of the coupling of quantum well space charge to its environment.     

Wigner-Boltzmann Monte Carlo approach to nanodevice simulation: from quantum to semiclassical transport

In this paper, we review and extend our recent works based on the Monte Carlo method to solve the Wigner-Boltzmann transport equation and model semiconductor nanodevices. After presenting the different possible approaches to quantum mechanical modelling, the formalism and the theoretical framework are described together with the particle Monte Carlo implementation using a technique fully compatible with semiclassical simulation. Examples are given to highlight the importance of considering both quantum and scattering effects in nanodevices operating at room temperature, such as resonant tunnelling diode (RTD), double-gate MOSFET and carbon nanotube FET. Quantum and semiclassical approaches are compared for transistor simulation. Finally, the phonon-induced electron decoherence in RTD and MOSFET is examined through the analysis of the density matrix elements computed from the Wigner function. This formalism is shown to be relevant for the quantitative analysis of devices operating in mixed quantum/semiclassical regime and to understand the transition between both regimes or between coherent and sequential tunnelling processes.     

Does Circulation in Individual Current States Survive in the Total Current Density?

We describe the two-dimensional simulation of a bent resonant tunneling diode structure which displays vortices in its total current density pattern over a range of applied bias. In contrast, a double gate n-MOSFET is shown where such circulation exists in individual subband states but does not survive in the total current density solution. Both devices are simulated assuming ballistic quantum transport in Si at 300 K.     

Microscopic Modelling of Quantum Open Systems: A Generalized Wigner-Function Approach

A microscopic approach for the simulation of quantum-transport phenomena in systems with open boundaries is proposed. We show that the application of the conventional Wigner-function formalism to this problem leads to unphysical results, like injection of coherent electronic states from the contacts. To overcome such basic limitation, we propose a generalization of the standard Wigner-function formulation, able to properly describe the incoherent nature of carrier injection at the device spatial boundaries. The latter constitutes a quantum mechanical derivation of the phenomenological injection model commonly employed in the simulation of open quantum devices.     

Bandgap nanoengineering of graphene tunnel diodes and tunnel transistors to control the negative differential resistance

By means of numerical simulation based on the Green’s function formalism on a tight binding Hamiltonian, we investigate different possibilities of achieving a strong effect of negative differential resistance in graphene tunnel diodes, the operation of which is controlled by the interband tunneling between both sides of the PN junction. We emphasize on different approaches of bandgap nanoengineering, in the form of nanoribbons (GNRs) or nanomeshes (GNMs), which can improve the device behaviour. In particular, by inserting a small or even zero bandgap section in the transition region separating the doped sides of the junction, the peak current and the peak-to-valley ratio (PVR) are shown to be strongly enhanced and weakly sensitive to the length fluctuations of the transition region, which is an important point regarding applications. The study is extended to the tunneling FET which offers the additional possibility of modulating the interband tunneling and the PVR. The overall work suggests the high potential of GNM lattices for designing high performance devices for either analog or digital applications.     

Scaling dependence of electron transport in nano-scale Schottky barrier MOSFETs

The scaling dependence of electron transport in the double-gated Schottky barrier MOSFET (DG-SBT) below 10 nm is investigated in the framework of quantum transport theory, using non-equilibrium Green’s function method. Simulation results show that the current-voltage characteristics in ultra-small DG-SBT are characterized by both resonant and direct tunneling effects. The electron potential in the 10-nm-scale DG-SBT surrounded by Schottky barriers acts as a resonant cavity and produce a negative differential resistance due to resonant tunneling effect. While, further scaling shallows the depth of the cavity and makes it difficult to form resonance levels. Hence, at the scaling limit, direct tunneling currents simply dominate the current-voltage characteristics of DG-SBT.     

Effective Mass Approach for n-MOSFETs on Arbitrarily Oriented Wafers

The general theory for quantum simulation of cubic semiconductor n-MOSFETs is presented within the effective mass equation approach. The full three-dimensional transport problem is described in terms of coupled transverse subband modes which arise due to quantum confinement along the body thickness direction. Couplings among the subbands are generated for two reasons: due to spatial variations of the confinement potential along the transport direction, and due to non-alignment of the device coordinate system with the principal axes of the constant energy conduction band ellipsoids. The problem simplifies considerably if the electrostatic potential is separable along transport and confinement directions, and further if the potential variations along the transport direction are slow enough to prevent dipolar coupling (Zener tunneling) between subbands. In this limit, the transport problem can be solved by employing two unitary operators to transform an arbitrarily oriented constant energy ellipsoid into a regular ellipsoid with principal axes along the transport, width and confinement directions of the device.     

Quantum Transport Simulation of Carrier Capture and Transport within Tunnel Injection Lasers

Hot electron distributions within the active region of quantum well lasers lead to gain suppression, reduced quantum efficiency, and increased diffusion capacitance, greater low-frequency roll-off and high-frequency chirp. Recently, tunnel injection lasers have been developed to minimize electron heating within the active quantum well region by direct injection of cool electrons from the separate confinement region into the lasing subband(s) through a tunneling barrier. Tunnel injection lasers, however, also present a rich physics of transport and scattering, and a correspondingly rich set of challenges to simulation and device optimization. For example, a Golden-Rule-based analysis of the carrier injection into the active region of the ideal tunnel injection laser would suggest approximately uniform injection of electrons among the nominally degenerate ground quantum well states from the separate confinement region states. However, such an analysis ignores (via a random-phase approximation among the final states) the basic real-space transport requirement that injected carriers still must pass through the wells sequentially, coherently or otherwise, with an associated attenuation of the injected current into each subsequent well due to electron-hole recombination in the prior well. Transport among the wells then can be either thermionic, or, of theoretically increasing importance for low temperature carriers, via tunneling. Coherent resonant tunneling between wells, however, is sensitive to the potential drops between wells that split the energies of the lasing subbands and (further) localizes the electron states to individual wells. In this work such transport issues are elucidated using Schrödinger Equation Monte Carlo (SEMC) based quantum transport simulation.     

Semiclassical and Quantum Transport in Si/SiO2 Superlattices

In the last few years, many research groups have been trying to develop electroluminescent devices based on silicon. In particular, it has been shown that low-dimensional structures, such as silicon clusters, quantum wires and quantum wells, are suitable for this purpose. In this work we investigate transport properties of a particular superlattice using two approaches. The first method is a Monte Carlo simulation of electron transport in the biased superlattice. The band structure is calculated using the envelope function approximation, and the scattering mechanisms introduced in the simulator are confined optical phonons. Owing to the particularly flat band structure, drift velocities are very low, but it will be shown that a parallel component of the electric field can significantly increase the vertical drift velocity. Moreover, a superlattice based device is proposed in order to obtain high recombination efficiency. Finally, a quantum calculation is introduced, in order to describe with higher accuracy the high field transport regime.