Wim Magnus

University of Antwerp | UA · Department of Physics

Today, further downscaling of mobile electronic devices poses serious problems, such as energy consumption and local heat dissipation. In this context, spin wave majority gates made of very thin ferromagnetic films may offer a viable alternative. However, similar downscaling of magnetic thin films eventually enforces the latter to operate as quasi-two dimensional magnets, the magnetic properties of which are not yet fully understood, especially those related to anisotropies and external magnetic fields in arbitrary directions. To this end, we have investigated the behaviour of an easy-plane and easy-axis anisotropic ferromagnet -- both in two and three dimensions -- subjected to a uniform magnetic field, applied along an arbitrary direction. In this paper, a spin-1/2 Heisenberg Hamiltonian with anisotropic exchange interactions is solved using double-time temperature-dependent Green's functions and the Tyablikov decoupling approximation. We determine various magnetic properties such as the Curie temperature and the magnetization as a function of temperature and the applied magnetic field, discussing the impact of the system's dimensionality and the type of anisotropy. The magnetic reorientation transition taking place in anisotropic Heisenberg ferromagnets is studied in detail. Importantly, spontaneous magnetization is found to be absent for easy-plane two-dimensional spin systems with short range interactions.

It is generally understood that the resistivity of metal thin films scales with film thickness mainly due to grain boundary and boundary surface scattering. Recently, several experiments and ab initio simulations have demonstrated the impact of crystal orientation on resistivity scaling. The crystal orientation cannot be captured by the commonly used resistivity scaling models and a qualitative understanding of its impact is currently lacking. In this work, we derive a resistivity scaling model that captures grain boundary and boundary surface scattering as well as the anisotropy of the band structure. The model is applied to Cu and Ru thin films, whose conduction bands are (quasi-)isotropic and anisotropic respectively. After calibrating the anisotropy with ab initio simulations, the resistivity scaling models are compared to experimental resistivity data and a renormalization of the fitted grain boundary reflection coefficient can be identified for textured Ru.

The resistivity of metal thin films is generally understood to increase with decreasing film thickness due to increased boundary surface and grain boundary scattering, the latter being a direct consequence of the average grain size typically reducing for thinner films. Recently, several experiments and ab initio simulations have demonstrated a dependency of resistivity scaling on the crystal orientation of the film, particularly in the case of an anisotropic Fermi surface. This anisotropy cannot be captured by the commonly used Mayadas-Shatzkes resistivity scaling model, which adopts an isotropic effective mass approximation for the electrons. As a qualitative understanding of the impact of conduction band anisotropy is currently lacking, we have extended the Mayadas-Shatzes approach to account for grain boundary and boundary surface scattering as well as the anisotropy of the electronic structure. Recently, we calibrated the extended model with Fermi surfaces obtained from ab initio simulations and successfully applied the model to Cu and Ru thin films, with a nearly isotropic and anisotropic Fermi surface respectively (arXiv:1711.00796).

The Wigner-Liouville equation is reformulated using a spectral decomposition of the classical force field instead of the potential energy. The latter is shown to simplify the Wigner-Liouville kernel both conceptually and numerically as the spectral force Wigner-Liouville equation avoids the numerical evaluation of the highly oscillatory Wigner kernel which is nonlocal in both position and momentum. The quantum mechanical evolution is instead governed by a term local in space and non-local in momentum, where the non-locality in momentum has only a limited range. An interpretation of the time evolution in terms of two processes is presented; a classical evolution under the influence of the averaged driving field, and a probability-preserving quantum-mechanical generation and annihilation term. Using the inherent stability and reduced complexity, a direct deterministic numerical implementation using Chebyshev and Fourier pseudo-spectral methods is detailed. For the purpose of illustration, we present results for the time-evolution of a one-dimensional resonant tunneling diode driven out of equilibrium.

Superconductivity is a macroscopic coherent state exhibiting various quantum phenomena such as magnetic flux quantization. When a superconducting ring is placed in a magnetic field, a current flows to expel the field from the ring and to ensure that the enclosed flux is an integer multiple of h/(2|e|). Although the quantization of magnetic flux in ring structures is extensively studied in literature, the applied magnetic field is typically assumed to be homogeneous, implicitly implying an interplay between field expulsion and flux quantization. Here, we propose to decouple these two effects by employing an Aharonov-Bohm-like structure where the superconducting ring is threaded by a magnetic core (to which the applied field is confined). Although the magnetic field vanishes inside the ring, the formation of vortices takes place, corresponding to a change in the flux state of the ring. The time evolution of the density of superconducting electrons is studied using the time-dependent Ginzburg-Landau equations.

A modeling approach, based on an analytical solution of the semiclassical multi-subband Boltzmann transport equation, is presented to study resistivity scaling in metallic thin films and nanowires due to grain boundary and surface roughness scattering. While taking into account the detailed statistical properties of grains, roughness and barriers as well as the material band structure and quantum mechanical aspects of scattering and confinement, the model does not rely on phenomenological fitting parameters.

This work investigates energy filtering in nanowires, where pass and stopbands are obtained by including superlattices in the wire. When a pair of such superlattices is placed in series, each being controlled by a gate, it can act as a transistor where the (mis-)alignment of its minibands turns the device on (off). It is shown that, in the ballistic current-regime, the transition between the on and off state occurs in a narrow gate-bias range, giving rise to sub-60 mV per decade switching behavior.

Fermi's golden rule underpins the investigation of mobile carriers propagating through various solids, being a standard tool to calculate their scattering rates. As such, it provides a perturbative estimate under the implicit assumption that the effect of the interaction Hamiltonian which causes the scattering events is sufficiently small. To check the validity of this assumption, we present a general framework to derive simple validity criteria in order to assess whether the scattering rates can be trusted for the system under consideration, given its statistical properties such as average size, electron density, impurity density et cetera. We derive concrete validity criteria for metallic nanowires with conduction electrons populating a single parabolic band subjected to different elastic scattering mechanisms: impurities, grain boundaries and surface roughness.

Fermi’s golden rule is often invoked to obtain scattering rates due to imperfections for semi-classical transport in different condensed matter systems. As it is an estimate for relatively small perturbations, its validity depends on the system and imperfection properties under consideration. We present a formal way to obtain easy to handle validity criteria, based on general system parameters, e.g. system size and momentum of the electron states, and the statistical properties of the imperfections. The criteria can also be obtained with a simple set of Feynman rules and corresponding diagrams. We show concrete examples of validity criteria for electron transport in metallic nanowires with several elastic scattering mechanisms, e.g. point defect or grain boundary scattering. We observe realistic nanowire examples where the scattering rate appears to be valid but also cases where the criteria are clearly violated. The latter indicates that higher order effects come into play, such as electrons being trapped between grain boundaries or at a rough surface, which cannot be described using Fermi’s golden rule. The presented validity criteria are therefore very useful to check whether or not the transport properties predicted by a semi-classical transport simulation can be trusted.

Being a standard tool to calculate the scattering rates that enter the collision term of the Boltzmann transport equation, Fermi's golden rule underpins a semi-classical investigation of mobile carriers propagating through various solids. As such, it provides a perturbative estimate of the scattering rates under the implicit assumption that the effect of the interaction Hamiltonian which causes the scattering events is sufficiently small. In order to check the validity of this assumption, we have developed easy-to-handle criteria to assess whether the predicted transport properties can be trusted for the system under consideration. Depending on the system's statistical properties, such as average size, electron density, impurity density etc., the analysis underlying these criteria is carried out for, but not limited to the multi-subband Boltzmann transport equation describing electron transport in metallic nanowires subjected to several elastic scattering mechanisms.

A weakly coupled system of two crossed graphene nanoribbons exhibits direct tunneling due to the overlap of the wavefunctions of both ribbons. We apply the Bardeen transfer Hamiltonian formalism, using atomistic band structure calculations to account for the effect of the atomic structure on the tunneling process. The strong quantum-size confinement of the nanoribbons is mirrored by the one-dimensional character of the electronic structure, resulting in properties that differ significantly from the case of inter-layer tunneling, where tunneling occurs between bulk two-dimensional graphene sheets. The current-voltage characteristics of the inter-ribbon tunneling structures exhibit resonance, as well as stepwise increases in current. Both features are caused by the energetic alignment of one-dimensional peaks in the density-of-states of the ribbons. Resonant tunneling occurs if the sign of the curvature of the coupled energy bands is equal, whereas a step-like increase in the current occurs if the signs are opposite. Changing the doping modulates the onset-voltage of the effects as well as their magnitude. Doping through electrostatic gating makes these structures promising for application towards steep slope switching devices. Using the atomistic empirical pseudopotentials based Bardeen transfer Hamiltonian method, inter-ribbon tunneling can be studied for the whole range of two-dimensional materials, such as transition metal dichalcogenides. The effects of resonance and of step-like increases in the current we observe in graphene ribbons are also expected in ribbons made from these alternative two-dimensional materials, because these effects are manifestations of the one-dimensional character of the density-of-states.

Efficient quantum mechanical simulation of tunnel field-effect transistors (TFETs) is indispensable to allow for an optimal configuration identification. We therefore present a full-zone 15-band quantum mechanical solver based on the envelope function formalism and employing a spectral method to reduce computational complexity and handle spurious solutions. We demonstrate the versatility of the solver by simulating a 40 nm wide In0.53Ga0.47As lineTFET and comparing it to p-n-i-n configurations with various pocket and body thicknesses. We find that the lineTFET performance is not degraded compared to semi-classical simulations. Furthermore, we show that a suitably optimized p-n-i-n TFET can obtain similar performance to the lineTFET.

The tunneling current between two crossed graphene ribbons is described invoking the empirical pseudopotential approximation and the Bardeen transfer Hamiltonian method. Results indicate that the density of states is the most important factor determining the tunneling current between small (nm) ribbons. The quasi-one dimensional nature of graphene nanoribbons is shown to result in resonant tunneling.

Ando's model provides a rigorous quantum-mechanical framework for electron-surface roughness scattering, based on the detailed roughness structure. We apply this method to metallic nanowires and improve the model introducing surface roughness distribution functions on a finite domain with analytical expressions for the average surface roughness matrix elements. This approach is valid for any roughness size and extends beyond the commonly used Prange-Nee approximation. The resistivity scaling is obtained from the self-consistent relaxation time solution of the Boltzmann transport equation and is compared to Prange-Nee's approach and other known methods. The results show that a substantial drop in resistivity can be obtained for certain diameters by achieving a large momentum gap between Fermi level states with positive and negative momentum in the transport direction.

A self-consistent analytical solution of the multi-subband Boltzmann transport equation with collision term describing grain boundary and surface roughness scattering is presented to study the resistivity scaling in metal nanowires. The different scattering mechanisms and the influence of their statistical parameters are analyzed. Instead of a simple power law relating the height or width of a nanowire to its resistivity, the picture appears to be more complicated due to quantum-mechanical scattering and quantization effects, especially for surface roughness scattering.

Ando's surface roughness model is applied to metallic nanowires and extended beyond small roughness size and infinite barrier limit approximations for the wavefunction overlaps, such as the Prange-Nee approximation. Accurate and fast simulations can still be performed without invoking these overlap approximations by averaging over roughness profiles using finite domain distribution functions to obtain an analytic solution for the scattering rates. The simulations indicate that overlap approximations, while predicting a resistivity that agrees more or less with our novel approach, poorly estimate the underlying scattering rates. All methods show that a momentum gap between left- and right-moving electrons at the Fermi level, surpassing a critical momentum gap, gives rise to a substantial decrease in resistivity.

A carefully chosen heterostructure can significantly boost the performance of tunnel fieldeffect transistors (TFET). Modelling of these hetero- TFETs requires a quantum mechanical (QM) approach with an accurate band structure to allow for a correct description of band-to-band-tunneling. We have therefore developed a fully QM 2D solver, combining for the first time a full zone 15-band envelope function formalism with a spectral approach, including a heterostructure basis set transformation. Simulations of GaSb/InAs broken gap TFETs illustrate the wide body capabilities and transparant transmission analysis of the formalism.

The tunneling current between two crossed graphene ribbons is described invoking the empirical pseudopotential approximation and the Bardeen transfer Hamiltonian method. Results indicate that the density of states is the most important factor determining the tunneling current between small (˜nm) ribbons. The quasi-one dimensional nature of graphene nanoribbons is shown to result in resonant tunneling.

Fixing the number of particles $N$, the quantum canonical ensemble imposes a constraint on the occupation numbers of single-particle states. The constraint particularly hampers the systematic calculation of the partition function and any relevant thermodynamic expectation value for arbitrary $N$ since, unlike the case of the grand-canonical ensemble, traces in the $N$-particle Hilbert space fail to factorize into simple traces over single-particle states. In this paper we introduce a projection operator that enables a constraint-free computation of the partition function and its derived quantities, at the price of an angular or contour integration. Being applicable to both bosonic and fermionic non-interacting systems in arbitrary dimensions, the projection operator approach provides closed-form expressions for the partition function $Z_N$ and the Helmholtz free energy $F_{\! N}$ as well as for two- and four-point correlation functions. While appearing only as a secondary quantity in the present context, the chemical potential potential emerges as a by-product from the relation $\mu_N = F_{\! N+1} - F_{\! N}$, as illustrated for a two-dimensional fermion gas with $N$ ranging between 2 and 500.

The envelope function method traditionally employs a single basis set which, in practice, relates to a single material because the matrix elements are generally only known in a particular basis. In this work, we defined a basis function transformation to alleviate this restriction. The transformation is completely described by the known inter-band momentum matrix elements. The resulting envelope function equation can solve the electronic structure in lattice matched heterostructures without resorting to boundary conditions at the interface between materials, while all unit-cell averaged observables can be calculated as with the standard envelope function formalism. In the case of two coupled bands, this heterostructure formalism is equivalent to the standard formalism while taking position dependent matrix elements.

Traditionally, a direct numerical solution of the Wigner-Liouville (WL) equation has been plagued with high computational burden and instability inherent to the integration of the highly oscillatory Wigner potential kernel. We have developed a method based on the spectral decomposition of the force which recasts the WL equation into a manageable form. By removing one integral, this new form is computationally less demanding. Furthermore a damping term naturally appears which reduces the instability caused by the oscillatory terms. Finally, the new form is local in position as opposed to the original WL equation which is non-local in both position and momentum. The spectral force WL equation is interpreted as representing two processes; a classical evolution with a constant force, and a local quantum generation term with positive and negative contributions mediated by the spectral components of the force. This interpretation allows for a straightforward implementation using a finite difference scheme for the classical evolution coupled with direct evaluation of the discretized generation terms. We observe a good match between results obtained using our method and theoretical results.

We calculate the resistivity contribution of tilted grain boundaries with varying parameters in sub-10nm diameter metallic nanowires. The results have been obtained with the Boltzmann transport equation and Fermi's golden rule, retrieving correct state-dependent relaxation times. The standard approximation schemes for the relaxation times are shown to fail when grain boundary tilt is considered. Grain boundaries tilted under the same angle or randomly tilted induce a resistivity decrease.

We propose a simple quantum mechanical model describing the time dependent diffusion current between two fermion reservoirs that were initially disconnected and characterized by different densities or chemical potentials. The exact, analytical solution of the model yields the transient behavior of the coupled fermion systems evolving to a final steady state, whereas the long-time behavior is determined by a power law rather than by exponential decay. Similar results are obtained for the entropy production which is proportional to the diffusion current.

We study the resistivity scaling in nanometer-sized metallic wires due to surface roughness and grain-boundaries, currently the main cause of electron scattering in nanoscaled interconnects. The resistivity has been obtained with the Boltzmann transport equation, adopting the relaxation time approximation of the distribution function and the effective mass approximation for the conducting electrons. The relaxation times are calculated exactly, using Fermi's golden rule, resulting in a correct relaxation time for every sub-band state contributing to the transport. In general, the relaxation time strongly depends on the sub-band state, something that remained unclear with the methods of previous work. The resistivity scaling is obtained for different roughness and grain-boundary properties, showing large differences in scaling behavior and relaxation times. Our model clearly indicates that the resistivity is dominated by grain-boundary scattering, easily surpassing the surface roughness contribution by a factor of 10.

The Wigner-Liouville (WL) equation is well suited to describe electronic transport in semiconductor devices. In the effective mass approximation the one dimensional WL equation reads ∂/∂t f(x, p, t) + p/m ∂/∂x f(x, p, t)-1/h2 ∫ dp' W(x, p-p')f(x, p', t) = 0; (1) with the Wigner kernel given by W(x, p) = -i/2π ∫ dx' exp (-i px'/h) [V (x + x'/2)-V (x-x'/2)].(2) The Wigner kernel introduces a non-local interaction with the potential V(x), in accordance with quantum theory. Unfortunately, even for this simple interaction the mathematical form includes a highly oscillatory component (exp [-i p·x/h]) which impedes stable numerical implementation based on finite differences or finite elements.

The Wigner function approach to quantum transport is well suited for application to nanoscaled electronic devices. However, the Wigner-Liouville equation is often formulated within the framework of the effective mass approximation. As the envelope function formalism based on k.p theory offers a more accurate description of the band structure, we have expanded the electron field operators in the corresponding envelope functions and rederived the Wigner transport equation accordingly. We obtain a set of coupled envelope-Wigner functions which enable us also to treat band- to-band transitions (BTBT) within the Wigner formalism. This way, we can provide a rigorous quantum mechanical treatment of BTBT events in phase space. Finally, we have extended this approach to the classical Boltzmann transport equation which introduces BTBT by invoking additional coupling terms on top of the classical drift-diffusion instead of ad-hoc generation and recombination terms.

We developed an envelope function formalism capable of describing the electronic structure in lattice matched heterostructures. The formalism takes into account the different nature of the materials involved by using matrix elements in the basis formed by the solutions of their respective bulk Hamiltonian. A transformation between these basis sets has been devised to allow for expansion in one consistent and complete set. This transformation is described without full knowledge of the basis function, as this would defeat the purpose of the envelope function method. We employ only the known interband momentum matrix elements to obtain the transformation coefficients. With this method it is not only possible to describe the electronic structure in heterostructures in a more rigorous way, it is also possible to describe band-to-band transitions through these heterostructures. In particular, we studied the transmission coefficients through broken-gap heterostructure. A large discrepancy was found with the effective mass approach, which predicts full transmission at a certain energy. Our method correctly predicts additional reflections due to the interface betweeen the two materials.

Heterostructure tunnel field-effect transistors (HTFET) are promising candidates for low-power applications in future technology nodes, as they are predicted to offer high on-currents, combined with a sub-60 mV/dec subthreshold swing. However, the effects of important quantum mechanical phenomena like size confinement at the heterojunction are not well understood, due to the theoretical and computational difficulties in modeling realistic heterostructures. We therefore present a ballistic quantum transport formalism, combining a novel envelope function approach for semiconductor heterostructures with the multiband quantum transmitting boundary method, which we extend to 2D potentials. We demonstrate an implementation of a 2-band version of the formalism and apply it to study confinement in realistic heterostructure diodes and p-n-i-n HTFETs. For the diodes, both transmission probabilities and current densities are found to decrease with stronger confinement. For the p-n-i-n HTFETs, the improved gate control is found to counteract the deterioration due to confinement.

Group IV based tunnel field-effect transistors generally show lower on-current than III-V based devices because of the weaker phonon-assisted tunneling transitions in the group IV indirect bandgap materials. Direct tunneling in Ge, however, can be enhanced by strain engineering. In this work, we use a 30-band k · p method to calculate the band structure of biaxial tensile strained Ge and then extract the bandgaps and effective masses at Γ and L symmetry points in k-space, from which the parameters for the direct and indirect band-to-band tunneling (BTBT) models are determined. While transitions from the heavy and light hole valence bands to the conduction band edge at the L point are always bridged by phonon scattering, we highlight a new finding that only the light-hole-like valence band is strongly coupling to the conduction band at the Γ point even in the presence of strain based on the 30-band k · p analysis. By utilizing a Technology Computer Aided Design simulator equipped with the calculated band-to-band tunneling BTBT models, the electrical characteristics of tensile strained Ge point and line tunneling devices are self-consistently computed considering multiple dynamic nonlocal tunnel paths. The influence of field-induced quantum confinement on the tunneling onset is included. Our simulation predicts that an on-current up to 160 (260) μA/μm can be achieved along with on/off ratio > 106 for VDD = 0.5 V by the n-type (p-type) line tunneling device made of 2.5% biaxial tensile strained Ge.

The phonon-assisted band-to-band tunneling (BTBT) current has been computed for a cylindrical nanowire tunneling field-effect transistor (TFET) with an all-round gate covering the source region. Although we have considered relatively thick wires, i.e. diameters ranging between 5 and 8 nm, we found that BTBT is considerably affected by the carrier confinement in the radial direction. Therefore, a self-consistent solution of the Schrödinger and Poisson equations must be carried out. For the latter, we have implemented a non-linear variational principle based on the modified local density approximation taking into account non-parabolic corrections for both conduction and valence bands. Our findings show not only that the confinement effects in nanowire TFETs have a stronger impact on the onset voltage of the tunneling current in comparison with their planar counterparts but also that the value of the onset voltage is overestimated when the valence band nonparabolicity is ignored.

The tunneling current has been computed for a cylindrical nanowire tunneling field-effect transistor (TFET) with an all-round gate that covers the source region. Being the underlying mechanism, band-to-band tunneling, mediated by electron–phonon interaction, is pronouncedly affected by carrier confinement in the radial direction and, therefore, involves the self-consistent solution of the Schrödinger and Poisson equations. The latter has been accomplished by exploiting a non-linear variational principle within the framework of the modified local density approximation taking into account the nonparabolicity of both the valence band and conduction band in relatively thick wires. Moreover, while the effective-mass approximation might still provide a reasonable description of the conduction band in relatively thick wires, we have found that the nonparabolicity of the valence band needs to be included. As a major conclusion, it is observed that confinement effects in nanowire tunneling field-effect transistors have a stronger impact on the onset voltage of the tunneling current in comparison with planar TFETs. On the other hand, the value of the onset voltage is found to be overestimated when the valence band nonparabolicity is ignored.

The Zener tunneling current flowing through a biased, abrupt p–n junction embedded in a cylindrical silicon nanowire is calculated. As the band gap becomes indirect for sufficiently thick wires, Zener tunneling and its related transitions between the valence and conduction bands are mediated by short-wavelength phonons interacting with mobile electrons. Therefore, not only the high electric field governing the electrons in the space-charge region but also the transverse acoustic (TA) and transverse optical (TO) phonons have to be incorporated in the expression for the tunneling current. The latter is also affected by carrier confinement in the radial direction and therefore we have solved the Schrödinger and Poisson equations self-consistently within the effective mass approximation for both conduction and valence band electrons. We predict that the tunneling current exhibits a pronounced dependence on the wire radius, particularly in the high-bias regime.

A figure of merit I60 is proposed for sub-60 mV/decade devices as the highest current where the input characteristics exhibit a transition from sub- to super-60 mV/decade behavior. For sub-60 mV/decade devices to be competitive with metal-oxide-semiconductor field-effect devices, I60 has to be in the 1-10 μA/μm range. The best experimental tunnel field-effect transistors (TFETs) in the literature only have an I60 of 6×10−3 μA/μm but using theoretical simulations, we show that an I60 of up to 10 μA/μm should be attainable. It is proven that the Schottky barrier FET (SBFET) has a 60 mV/decade subthreshold swing limit while combining a SBFET and a TFET does improve performance.

In this chapter, the junctionless nanowire transistor is introduced as an alternative device concept to the inversion mode nanowire MOSFET. We first discuss the basic working principle based on an analytical model for long thick nanowires. Then, we scale down the wire radius and discuss the impact of size quantization on the lowfield mobility. Next, we also scale down the gate length to investigate the purely ballistic junctionless nanowire and its equivalence to the inversion mode MOSFET nanowire in this regime. Finally, we consider an advanced transport model to investigate the shortchannel junctionless nanowire including strain.

We propose a simple quantum mechanical model describing the time dependent diffusion current between two fermion reservoirs that were initially disconnected and characterized by different densities or chemical potentials. The exact, analytical solution of the model yields the transient behavior of the coupled fermion systems evolving to a final steady state, whereas the long-time behavior is determined by a power law rather than by exponential decay. Similar results are obtained for the entropy production which is proportional to the diffusion current.

This book covers one of the most important device architectures that have been widely researched to extend the transistor scaling: FinFET. Starting with theory, the book discusses the advantages and the integration challenges of this device architecture. It addresses in detail the topics such as high-density fin patterning, gate stack design, and source/drain engineering, which have been considered challenges for the integration of FinFETs. The book also addresses circuit-related aspects, including the impact of variability on SRAM design, ESD design, and high-T operation. It discusses a new device concept: the junctionless nanowire FET.

We investigate a promising tunnel FET configuration having a gate on the source only, which is simultaneously exhibiting a steeper subthreshold slope and a higher ON-current than the lateral tunneling configuration with a gate on the channel. Our analysis is performed based on a recently developed 2-D quantum-mechanical simulator calculating band-to-band tunneling and including quantum confinement (QC). It is shown that the two disadvantages of the structure, namely, the sensitivity to gate alignment and the physical oxide thickness, are mitigated by placing a counter-doped parallel pocket underneath the gate-source overlap. The pocket also significantly reduces the field-induced QC. The findings are illustrated with all-Si and all-Ge gate-onsource-only tunnel field-effect transistor simulations.

Starting from Feynman's Lagrangian description of quantum mechanics, we propose a method to construct explicitly the propagator for the Wigner distribution function of a single system. For general quadratic Lagrangians, only the classical phase space trajectory is found to contribute to the propagator. Inspired by Feynman's and Vernon's influence functional theory we extend the method to calculate the propagator for the reduced Wigner function of a system of interest coupled to an external system. Explicit expressions are obtained when the external system consists of a set of independent harmonic oscillators. As an example we calculate the propagator for the reduced Wigner function associated with the Caldeira-Legett model.

We develop a model for the tunnel field-effect transistor (TFET) based on the Wentzel-Kramer-Brillouin approximation which improves over existing semi-classical models employing generation rates. We hereby introduce the concept of a characteristic tunneling length in direct semiconductors. Based on the model, we show that a limited density of states results in an optimal doping concentration as well as an optimal material's band gap to obtain the highest TFET on-current at a given supply voltage. The observed optimal-doping trend is confirmed by 2-dimensional quantum-mechanical simulations for silicon and germanium.

The real time propagator of the Wigner distribution function can be constructed from the Wigner-Liouville equation as a phase space path integral. By analogy with the Feynman path integral one can define a new effective Lagrangian of the system in the Wigner-Weyl representation. The effects of gauge transformations and geometric constraints on the action are discussed. In particular we discuss the dynamics of a non-interacting 2DEG on a Hall strip.

The propagator of the Wigner function is constructed from the Wigner–Liouville equation as a phase space path integral over a new effective Lagrangian. In contrast to a paper by Barker and Murray (1983) [1], we show that the path integral can in general not be written as a linear superposition of classical phase space trajectories over a family of non-local forces. Instead, we adopt a saddle point expansion to show that the semiclassical Wigner function is a linear superposition of classical solutions for a different set of non-local time dependent forces. As shown by a simple example the specific form of the path integral makes the formulation ideal for Monte Carlo simulation.

In the realm of Ehrenfest's theorem, classical trajectories obeying Newton's laws have been proven useful to construct explicit solutions to the time-dependent Wigner-Liouville equation. Whereas previous works have particularly focused on the initial distribution function as a vehicle found to carry the signatures of quantum statistics into the time-dependent solution, the present paper shows that the Lagrange-Charpit method based on classical trajectories can be successfully invoked as well to tackle quantum mechanical features with no classical counterpart, such as tunneling. Newtonian trajectories can be used to solve both the Boltzmann and the Wigner-Liouville equations. In particular, tunneling can be treated with Newtonian trajectories. Numerical instabilities related to the Wigner kernel can be avoided by defining an appropriate effective potential.

We theoretically investigate the phonon, surface roughness and ionized impurity limited low-field mobility of ultrathin silicon n-type nanowire junctionless transistors in the long channel approximation with wire radii ranging from 2 to 5 nm, as function of gate voltage. We show that surface roughness scattering is negligible as long as the wire radius is not too small and ionized impurity scattering is the dominant scattering mechanism. We also show that there exists an optimal radius where the ionized impurity limited mobility exhibits a maximum.

A quantum mechanical procedure to calculate phonon-assisted tunneling current in indirect semiconductors in a two-dimensional structure is demonstrated. Applying the procedure to two types of double-gate silicon tunnel field-effect transistor (TFET) structures, it is observed that semiclassical predictions strongly overestimate the subthreshold swing and the device current. Furthermore, while the semiclassical simulation suggests a higher current for one of the investigated TFET devices, a proper quantum mechanical calculation comes to the opposite conclusion. This result, which is expected to apply to direct semiconductors as well, proves the importance of correct quantum mechanical tunneling models for performance predictions of novel devices such as TFETs.

Going beyond the existing semiclassical approach to calculate band-to-band tunneling (BTBT) current we have developed a quantum mechanical model incorporating confine- ment effects and multiple electron and hole valleys to calculate the tunnel current in a tunnel field-effect transistor. Comparison with existing semiclassical models reveals a big shift in the onset of tunneling due to energy quantization. We show that the big shift due to quantum confinement is slightly reduced by taking penetration into the gate dielectric into account. We further propose a modified semiclassical model capable of accounting for quantum confinement.

A general framework to calculate the Zener current in an indirect semiconductor with an externally applied potential is provided. Assuming a parabolic valence and conduction band dispersion, the semiconductor is in equilibrium in the presence of the external field as long as the electron-phonon interaction is absent. The linear response to the electron-phonon interaction results in a non-equilibrium system. The Zener tunneling current is calculated from the number of electrons making the transition from valence to conduction band per unit time. A convenient expression based on the single particle spectral functions is provided, enabling the evaluation of the Zener tunneling current under any three-dimensional potential profile. For a one-dimensional potential profile an analytical expression is obtained for the current in a bulk semiconductor, a semiconductor under uniform field, and a semiconductor under a non-uniform field using the WKB (Wentzel–Kramers–Brillouin) approximation. The obtained results agree with the Kane result in the low field limit. A numerical example for abrupt p-n diodes with different doping concentrations is given, from which it can be seen that the uniform field model is a better approximation than the WKB model, but a direct numerical treatment is required for low bias conditions.

For the Si-based TFET to beat the MOSFET performance and allow ultra-low voltage operation with re-use of a lot of the existing processing expertise, critical device optimization is needed whereby a combination of several performance boosters must be implemented. Heterostructures and an appropriate stress profile are necessary requirements. The largest design impact is expected from scaling the effective oxide thickness and the body thickness. Field-induced quantum confinement affects most theoretical predictions today and needs to be addressed in the design optimization. Overall, there are still significant challenges both in modeling, processing and characterization of the device. Progress in all three areas is required to uncover the full potential of the TFET.

In this work, cylindrical junctionless nanowire pinch-off FETs with a circular horizontal cross-section are simulated. Advanced simulation methods based on the self consistent solution of the Poisson equation (PE) and the 6x6 k · p Schrodinger equation (SE) (for pFETs) or the effective mass SE (for nFETs) are employed allowing us to handle quantum confinement, stress/strain, and arbitrary crystallographic orientations. Using these advanced simulation methods the change of the electrostatics in junctionless nanowire pinch-off FETs is studied when strain and/or an arbitrary crystallographic orientation is considered. In this work we consider nanowire FETs with an infinitely long channel such that the Si body of the device is homogeneous, i.e. it is taken to be translation invariant in the channel direction (z). The simulation results for both homogeneous channel nFETs and pFETs are presented.

Being the working principle of a tunnel field-effect transistor, band-to-band tunneling is given a rigorous quantum mechanical treatment to incorporate confinement effects, multiple electron and hole valleys, and interactions with phonons. The model reveals that the strong band bending near the gate dielectric, required to create short tunnel paths, results in quantization of the energy bands. Comparison with semiclassical models reveals a big shift in the onset of tunneling. The effective mass difference of the distinct valleys is found to reduce the subthreshold swing steepness.

Several years ago, a novel device concept was proposed : the nanowire (NW) iJFET [1]. Today, this device concept is being explored by several research teams [1-3] and is also known as the pinch-off FET (POFET) or junctionless transistor. The most important advantage of the junctionless transistor is the uniform doping throughout source, channel and drain which greatly simplifies its fabrication. We have performed modeling and simulations to compare the performance of the junctionless pinch-off FET with that of inversion mode devices. In order to make the comparison, we address the regime of thick and long nanowires through analytical modeling of the current- voltage characteristics, while for long and thin nanowires we perform dissipative transport modelling to obtain the low-field mobility. Finally, ballistic transport modelling is performed using the sub band decomposition method for ultra- short nanowires. [4pt] [1] B. Sor'ee, et al., Journal of Computational Electronics, vol.7, issue 3, 380-383 2008. [0pt] [2] B. Sor'ee, et al., Nanoelectronics days 2010, Aachen, Germany. [0pt] [3] J.-P. Colinge, et al., Nature Nanotechnology 5, 225-229, 2010

In previous studies, we showed that the classical equations of motion provide a solution to quantum dynamics, if appropriately incorporated in the Wigner distribution function, exactly reformulated in a type of Boltzmann equation. However, this earlier work was limited to scalar potentials. In the presence of an electromagnetic field, we now show that this description in terms of classical paths remains valid, despite the fact that the definition of the Wigner distribution function is not gauge invariant. Some analytical results are also presented.

With the scaling in the semiconductor device dimensions, Zener tunneling has become an important source of leakage in conventional MOSFET devices but it could also provide drive current for a novel type of tunnel transistor. A good understanding of the process of Zener tunneling is therefore required and present-day one-dimensional semi-classical models fall short of explaining tunneling in devices with potential profiles with a pronounced two-dimensional shape. We have developed a formalism to calculate the phonon-assisted current under a given three dimensional external potential profile. The current is calculated from the transition probability for an electron to go from the valence to the conduction band. The transition probability is determined from the spectral functions corresponding to the valence and the conduction band. In the presence of a one-dimensional uniform low electric field, the Kane model is recovered. An example of the formalism is given for the case of an abrupt p-n diode and compared with existing semi-classical models. It is seen that the uniform field model is actually better than the WKB model but that none of the semi-classical models give good results at low bias conditions.

We calculate the electron mobility for a metal-oxide-semiconductor system with a metallic gate, high- κ dielectric layer, and III-V substrate, including scattering with longitudinal-optical (LO) polar-phonons of the III-V substrate and with the interfacial excitations resulting from the coupling of insulator and substrate optical modes among themselves and with substrate plasmons. In treating scattering with the substrate LO-modes, multisubband dynamic screening is included and compared to the dielectric screening in the static limit and with the commonly used screening model obtained by defining an effective screening wave vector. The electron mobility components limited by substrate LO phonons and interfacial modes are calculated for In _0.53 Ga _0.47 As and GaAs substrates with SiO ₂ and HfO ₂ gate dielectrics. The mobility components limited by the LO-modes and interfacial phonons are also investigated as a function of temperature. Scattering with surface roughness, fixed interface charge, and nonpolar-phonons is also included to judge the relative impact of each scattering mechanism in the total mobility for In _0.53 Ga _0.47 As with HfO ₂ gate dielectric. We show that InGaAs is affected by interfacial-phonon scattering to an extent larger than Si, lowering the expected performance, but probably not enough to question the technological relevance of InGaAs.

Since Ehrenfest’s theorem, the role and importance of classical paths in quantum dynamics have been examined by several means. Along this line, we show that the classical equations of motion provide a solution to quantum dynamics, if appropriately incorporated into the Wigner distribution function, exactly reformulated in a type of Boltzmann equation. Also the quantum-mechanical features of the canonical ensemble can be studied in this framework of Newtonian dynamics, if the initial distribution function is appropriately constructed from the statistical operator.

In order to simplify the numerical investigation of carrier transport in nanodevices without jeopardizing the rigor of a full quantum mechanical treatment, we have exploited an existing variational principle to solve self-consistently Poisson's equation and Schrödinger's equation as well as an appropriate transport equation within the scope of the generalized local density approximation (GLDA). In this work, as a benchmark, we have applied our approach to compute the ballistic current density and electron concentration in a Si nanowire.

A simplified quantum mechanical model is developed to investigate quantum transport features such as the electron concentration and the current flowing through a silicon nanowire metal-oxide-semiconductor field-effect transistor (MOSFET). In particular, the electron concentration is extracted from a self-consistent solution of the Schrödinger and Poisson equations as well as the ballistic Boltzmann equation which have been solved by exploiting a nonlinear variational principle within the framework of the generalized local density approximation. A suitable action functional has been minimized and details of the implementation and its numerical minimization are given. The current density and its related current-voltage characteristics are calculated from the one-dimensional ballistic steady-state Boltzmann transport equation which is solved analytically by using the method of characteristic curves. The straightforward implementation, the computational speed and the good qualitative behavior of the transport characteristics observed in our approach make it a promising simulation method for modeling quantum transport in nanowire MOSFETs.

Recently, a renewed interest in Zener tunneling has arisen because of its increasing impact on semiconductor device performance at nanometer dimensions. In this paper we evaluate the tunnel probability under the action of a nonuniform electric field using a two-band model and arrive at significant deviations from the commonly used Kane’s model, valid for weak uniform fields only. A threshold on the junction bias where Kane’s model for Zener tunneling breaks down is determined. Comparison with Kane’s model particularly shows that our calculation yields a higher tunnel probability for intermediate electric fields and a lower tunnel probability for high electric fields. When performing a current calculation comparing to the WKB approximation for the case of an abrupt p-n junction significant differences concerning the shape of the I-V curve are demonstrated.

The use of high mobility channel materials such as Ge and III/V compounds for CMOS applications is being explored. The introduction of these new materials also opens the path towards the introduction of novel device structures which can be used to lower the supply voltage and reduce the power consumption. The results illustrate the possibilities that are created by the combination of new materials and devices to allow scaling of nanoelectronics beyond the Si roadmap.

We studied the optical Aharonov-Bohm effect for an exciton in a semiconductor quantum ring. A perpendicular electric field applied to a quantum ring with large height, is able to tune the exciton ground state energy such that it exhibits a weak observable Aharonov-Bohm oscillations. This Aharonov-Bohm effect is tunable in strength and period.

We explain the basic operation of a nanowire pinch-off FET and graphene nanoribbon tunnelFET. For the nanowire pinch-off FET we construct an analytical model to obtain the threshold voltage as a function of radius and doping density. We use the gradual channel approximation to calculate the current-voltage characteristics of this device and we show that the nanowire pinch-off FET has a subthreshold slope of 60 mV/dec and good ION and ION/IOFF ratios. For the graphene nanoribbon tunnelFET we show that an improved analytical model yields more realistic results for the transmission probability and hence the tunneling current. The first simulation results for the graphene nanoribbon tunnelFET show promising subthreshold slopes.

With the introduction of ever smaller dimensions in modern day semiconductor devices, Zener tunneling can no longer be neglected and is affecting device performance. On one hand Zener tunneling is responsible for a leakage current in classical devices such as the MOSFET, on the other hand it provides the drive current for some new devices under investigation such as the tunnel field-effect transistor (TFET). A popular model to calculate the Zener tunneling probability is the Kane model but this model is only valid for Zener tunneling in weak uniform fields. The Kane model can be extended to the case of a non-uniform field using a WKB approximation. But the proper way to calculate the tunneling probability is to derive it directly from the transmission probability of an incoming electron. Using a two-band model we compare the results of a direct calculation of the transmission probability with that calculated using the WKB approximation or the Kane model. We conclude that the Kane model breaks down in the case of high fields and low junction bias.

For many decades the Boltzmann distribution function has been used to calculate the non-equilibrium properties of mobile particles undergoing the combined action of various scattering mechanisms and externally applied force fields. When the latter give rise to the occurrence of inhomogeneous potential profiles across the region through which the particles are moving, the numerical solution of the Boltzmann equation becomes a highly complicated task. In this work we highlight a particular algorithm that can be used to solve the time dependent Boltzmann equation as well as its quantum mechanical extension, the Wigner–Boltzmann equation. As an illustration, we show the calculated distribution function describing electrons propagating under the action of both a uniform and a pronouncedly non-uniform electric field.

We present comprehensive calculations of the low-field hole mobility in Ge p -channel inversion layers with SiO ₂ insulator using a six-band k ∙ p band-structure model. The cases of relaxed, biaxially, and uniaxially (both tensily and compressively) strained Ge are studied employing an efficient self-consistent method—making use of a nonuniform spatial mesh and of the Broyden second method—to solve the coupled envelope-wave function k ∙ p and Poisson equations. The hole mobility is computed using the Kubo–Greenwood formalism accounting for nonpolar hole-phonon scattering and scattering with interfacial roughness. Different approximations to handle dielectric screening are also investigated. As our main result, we find a large enhancement (up to a factor of 10 with respect to Si) of the mobility in the case of uniaxial compressive stress similarly to the well-known case of Si. Comparison with experimental data shows overall qualitative agreement but with significant deviations due mainly to the unknown morphology of the rough Ge-insulator interface, to additional scattering with surface optical phonon from the high- κ insulator, to Coulomb scattering interface traps or oxide charges—ignored in our calculations—and to different channel structures employed.

The dynamics of electrons and holes propagating through the nano-scaled channels of modern semiconductor devices can be seen as a widespread manifestation of non-equilibrium statistical physics and its ruling principles. In this respect both the devices that are pushing conventional CMOS technology towards the final frontiers of Moore’s law and the upcoming set of alternative, novel nanostructures grounded on entirely new concepts and working principles, provide an almost unlimited playground for assessing physical models and numerical techniques emerging from classical and quantum mechanical non-equilibrium theory. In this paper we revisit the Boltzmann as well as the Wigner–Boltzmann equation which offers a valuable platform to study transport of charge carriers taking part in drive currents. We focus on a numerical procedure that regained attention recently as an alternative tool to solve the time-dependent Boltzmann equation for inhomogeneous systems, such as the channel regions of field-effect transistors, and we discuss its extension to the Wigner–Boltzmann equation. Furthermore, we pay attention to the calculation of tunneling leakage currents. The latter typically occurs in nano-scaled transistors when part of the carrier distribution sustaining the drive current is found to tunnel into the gate due the presence of an ultra-thin insulating barrier separating the gate from the channel region. In particular, we discuss the paradox related to the very existence of leakage currents established by electrons occupying quasi-bound states, while the (real) wave functions of the latter cannot carry net currents. Finally, we describe a simple model to resolve the paradox as well as to estimate gate currents provided the local carrier generation rates largely exceed the tunneling rates.

Significant research efforts are devoted to the development of alternative high dielectric constant oxides (high-k) to sustain the requirements of the next generations of DRAM and FLASH memories. For the metal-insulator-metal applications, the identification of an oxide with a dielectric constant higher than 50 and a crystallization temperature lower than 650 C is a corner stone in the selection of the materials; while for non volatile memory (NVM) applications, an oxide with a dielectric constant ranging between 9 and 30, a large band gap (> 6 eV) are targeted. While significant efforts have been dedicated to the integration of possible material candidates, little is known on their intrinsic properties. Through this paper, we provide an overview of the parameters that determine the electronic and the polarizability response of tantalates, niobiates and rare-earth based scandates oxides eligible for the development of the next generation of memory based applications.

In this study we investigate the effect of device topology on the ballistic current in n-channel metal-oxide-semiconductor field-effect transistors. Comparison of the nanoscale planar and double-gate devices reveals that, down to a certain thickness of the double gate film, the ballistic current flowing in the double gate device is twice as large compared to its planar counterpart. On the other hand, further thinning of the film beyond this threshold is found to change noticeably the confinement and transport characteristics, which are strongly depending on the film material and the surface orientation. For double gate Ge and Si devices there exists a critical film thickness below which the transverse gate field is no longer effectively screened by the inversion layer electron gas and mutual inversion of the two gates is turned on. In the case of GaAs and other similar III-V compounds, a decrease in the film thickness may drastically change the occupation of the L-valleys and therefore amend the transport properties. The simulation results show that, in both cases, the ballistic current and the transconductance are considerably enhanced.

In this paper we investigate the basic physics of charge carriers (electrons) leaking out of the inversion layer of a metal-oxide-semiconductor capacitor with a biased gate. In particular, we treat the gate leakage current as resulting from two combined processes: (1) the time-dependent decay of electron wave packets representing the inversion-layer charge and (2) the local generation of “new” electrons replacing those that have leaked away. As a result, the gate current simply emerges as the ratio of the total charge in the inversion layer to the tunneling lifetime. The latter is extracted from the quantum dynamics of the decaying wave packets, while the generation rate is incorporated as a phenomenological source term in the continuity equation. Not only do the gate currents calculated with this model agree very well with experiment, the model also provides an onset to solve the paradox of the current-free bound states representing the resonances of the Schrödinger equation that governs the fully coupled metal-oxide-semiconductor system.

The carbon nanotube and graphene nanoribbon band structure is derived using the two-band k · p method and shown to have a similar band structure as a III-V semiconductor. Contrary to a previous claim, it is shown that the tunnelling probability is lower for a graphene based semiconductor than for a III-V semiconductor with the same bandgap. Considering the relation between the bandgap and the effective mass we conclude that a graphene based semiconductor is not well-suited for a classical device which suffers from Zener tunnelling, but is rather promising for a device which has its working principle based on Zener tunnelling.

In principle, transport of charged carriers in nanometer sized solid-state devices can be fully characterized once the non-equilibrium distribution function describing the carrier ensemble is known. In this light, we have revisited the Boltzmann and the Wigner distribution functions and the framework in which they emerge from the classical respectively quantum mechanical Liouville equation. We have assessed the method of the characteristic curves as a potential workhorse to solve the time dependent Boltzmann equation for carriers propagating through spatially non-uniform systems, such as nanodevices. In order to validate the proposed solution strategy, we numerically solve the Boltzmann equation for a one-dimensional conductor mimicking the basic features of a biased low-dimensional transistor operating in the on-state. Finally, we propose a computational scheme capable of extending the benefits of the above mentioned solution strategy when it comes to solve the Wigner–Liouville equation. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

An analytical model has been constructed for a silicon nanowire pinch-off FET. This is a uniformly doped nanowire with surrounding oxide and gate. For large radii nanowires, in the long channel approximation, we obtain an analytical expression for the depletion length in the radial direction. Making use of the gradual channel approximation, we are able to investigate the current both above and below threshold. The results obtained from the analytical model are compared with the self-consistent numerical solutions of a self-consistent Poisson solver. We show that the subthreshold slope is 60 mV/dec and good I_ON/I_OFF ratios and I_ON values are obtained for this device.

A superconducting transformer is implemented using standard Si processing steps and operated with DC current signals to form a quantized logic switch. Two discrete magnetic flux states are realized in well-controlled manner and switched between, defining a two-level system. Related material and transport characteristics are reported. Based on these results, we propose using multi-transformer arrays for possible future quantum information processing trial.

The effect of a metallic gate on the bound states of a shallow donor located near the gate is studied. We calculate the energy spectrum as a function of the distance between the metallic gate and the donor and find an anti-crossing behavior in the energy levels for certain distances. We show how a transverse electric field can tune the average position of the electron with respect to the metallic gate and the impurity.

We studied the coupled impurity states in a freestanding semiconductor nanowire (NW), within the effective mass approximation and including the effect of the dielectric mismatch, by using finite element method. Bonding and anti-bonding states are found and their energies converge with increasing distance di between the two impurities. The dependence of the binding energy on the wire radius R and the distance di between the two impurities is investigated, and we compare it with the result of a freestanding NW that contains a single impurity.

In principle, transport of charged carriers in nanometer sized solid-state devices can be fully characterized once the non- equilibrium distribution function describing the carrier ensemble is known. In this light, we have revisited the Boltzmann and the Wigner distribution functions and the framework in which they emerge from the classical respectively quantum mechanical Liouville equation. We have assessed the method of the characteristic curves as a potential workhorse to solve the time dependent Boltzmann equation for carriers propagating through spatially non-uniform systems, such as nanodevices. In order to validate the proposed solution strategy, we numerically solve the Boltzmann equation for a one- dimensional conductor mimicking the basic features of a biased low-dimensional transistor operating in the on-state. Finally, we propose a computational scheme capable of extending the benefits of the above mentioned solution strategy when it comes to solve the Wigner-Liouville equation.

We demonstrate a simple and robust method for inducing and detecting changes of magnetic flux quantization in the absence of an externally applied magnetic field. In our device, an isolated ring is interconnected with two access loops via permalloy cores, forming a superconducting transformer. By applying and tuning a direct current at the first access loop, the number of flux quanta trapped in the isolated ring is modified without the aid of an external field. The flux state of the isolated ring is simply detected by recording the evolution of the critical current of the second access loop.

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.

The tunnel field-effect transistor (TFET) is a promising candidate for the succession of the MOSFET at nanometer dimensions. In general, the TFET current can be decomposed into two components referred to as point tunneling and line tunneling. In this paper we derive a compact analytical model for the current due to point tunneling complementing the previously derived analytical model for line tunneling. We show that the derived analytical expression for point tunneling provides a more consistent estimate of the TFET current than a commercial device simulator. Both the line and point tunneling current do not show a fixed subthreshold-slope. Three key parameters for design of a TFET are: bandgap, dielectric thickness and source doping level. A small bandgap is beneficial for a high TFET on-current and a low onset voltage. Point tunneling and line tunneling show a strong dependance on gate dielectric thickness and doping concentration respectively.

We derive an analytical model for the electrostatics and the drive current in a silicon nanowire operating in JFET mode. We show that there exists a range of nanowire radii and doping densities for which the nanowire JFET satisfies reasonable device characteristics. For thin nanowires we have developed a self-consistent quantum mechanical model to obtain the electronic structure.

We investigated the energy levels of excitons (X) and trions (X ±) in free-standing nanowires with strong lateral carrier confinement. Within the adiabatic approximation the two (resp. three) particle problem reduces to an effective one (resp. two) dimensional Schrödinger equation for the relative motion which is solved numerically. Dielectric mismatch effects are taken into account that result in a distorted Coulomb interaction between the charged particles. We obtain the X and X ± binding energies as function of the dielectric mismatch ratio. To obtain these energies, we constructed tractable analytical expressions for the effective potential in the wire which significantly reduces the amount of computational time.

Quantum mechanical features of the electron transport in a SOI MOSFET are described within the Wigner function formalism which explicitly deals with electron scattering due to ionized impurities, acoustic phonons and surface roughness at the Si/SiO2 interface. The calculated device characteristics are obtained as a function of the thickness of the semiconductor layer. An analysis of the I–V characteristics of the MOSFET shows a substantial reduction of the short-channel effect with a decrease in the channel thickness of the device.

The tunnel field-effect transistor (TFET) is a promising candidate for the succession of the MOSFET at nanometer dimensions. Due to the absence of a simple analytical model for the TFET, the working principle is generally not well understood. In this paper a new TFET structure is introduced and using Kanepsilas model, an analytical expression for the current through the TFET is derived. Furthermore, a compact expression for the TFET current is derived and conclusions concerning TFET design are drawn. The obtained analytical expressions are compared with results from a 2D device simulator and good agreement at low gate voltages is demonstrated.

We investigated the energy levels of excitons (X) and trions (X±) in free-standing nanowires with strong lateral carrier confinement. Within the adiabatic approximation the two (resp. three) particle problem reduces to an effective one (resp. two) dimensional Schrödinger equation for the relative motion which is solved numerically. Dielectric mismatch effects are taken into account that result in a distorted Coulomb interaction between the charged particles. We obtain the X and X± binding energies as function of the dielectric mismatch ratio. To obtain these energies, we constructed tractable analytical expressions for the effective potential in the wire which significantly reduces the amount of computational time. Furthermore, X and X± binding energies as function of the wire radius were calculated and fitted. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Carbon nanotubes (CNTs) are a promising candidate to replace copper interconnects. An ab initio study is presented on the conductance of a closed-packed bundle of very narrow metallic (4,0) CNTs, which is vertically placed on a Cu (100) surface. The intertube interactions have no significant impact on the conductance. The conductance is highly dependent on the exact geometry of the interface, which is varying between 0.6 and 1.8 conductance quanta, while the theoretical maximum of the CNT is three conductance quanta. The wave-function interference can lead to conductance suppression when the packing is too high. Both features are explained by using an orbital picture.

We investigate the energy levels of excitons (X) and trions (X±) in free-standing nanowires with strong lateral carrier confinement. Within the adiabatic approximation the two (resp. three) particle problem reduces to an effective one (resp. two) dimensional Schrödinger equation for the relative motion which is solved numerically. Dielectric mismatch effects are taken into account that result in a distorted Coulomb interaction between the charged particles. We obtain the X and X± binding energies as function of the dielectric mismatch ratio. We also construct tractable analytical expressions for the effective potential which significantly reduce the amount of computational time for future work.

We have investigated the effect of a metallic gate on the bound states of a shallow donor located near the gate. We calculate the energy spectrum as a function of the distance between the metallic gate and the donor and find an anticrossing behavior in the energy levels for certain distances. We show how a transverse electric field can tune the average position of the electron with respect to the metallic gate and the impurity.

Shallow impurity states in a freestanding semiconductor nanowire and in a semiconductor nanowire surrounded by a metallic gate are studied within the effective mass approximation. We calculate the total energy of the electron and the binding energy by using (1) a variational approach, which provides an upper bound to the electron energy, and (2) the finite element method which is “numerically” exact. The dependence of the binding energy and the extent of the shallow impurity wave function on the wire radius R and the ionized impurity position in the nanowire is examined. The validity of the often used variational calculation is critically examined by calculating the difference of the binding energies obtained from the two different methods as a function of the wire radius R and the ionized impurity position.

The cylindrical geometry of nanowire surrounding gate MOSFETs gives rise to outstanding electrostatic control in comparison to planar devices. On the other hand, we expect that for ultrasmall nanowire diameters, the interaction of electrons with the surface (e.g. surface roughness and high-k) will be detrimental for device performance due to mobility degradation. In order to avoid these surface interactions we consider a surrounding gate nanowire operated not in MOSFET mode, but in ``JFET mode.'' We thus consider a nanowire with silicon body radius R and surrounding oxide of thickness tox with a surrounding metal gate where both source, drain and silicon body are doped uniformly with a donor density ND. Applying a negative gate voltage pushes the electrons away from the interface between the insulator and metal gate, and as a result a depletion region is induced. For sufficient negative gate voltage the depletion region reaches the center of the silicon body, and pinch-off occurs. For large radii, we construct a compact model, and we show that reasonable pinch- off voltages are realized when the wire radius or the donor density is sufficiently small. Using the gradual channel approximation we are able to obtain current-voltage characteristics that constitute a ``proof of concept'' for this device. In the case of ultrasmall radii, we perform a quantum mechanical analysis of the electronic structure.

A robust algorithm to get the chemical potential of the particle reservoirs for the self consistent full 2D Schrödinger-Poisson solver is proposed. Using this algorithm we study the effect of junction depth on ballistic current. Simulation results show that shallow junctions come with much better on to off current ratio while it keeps the on-state transconductance at the same level as the deeper junction device.

We investigated the lowest energy levels of trions (charged excitons) in freestanding nanowires with strong lateral carrier confinement. Within the adiabatic approximation, the three-particle problem reduces to an effective two-dimensional Schrödinger equation for the relative motion which is solved numerically. Dielectric mismatch effects are taken into account, which results in a distorted Coulomb interaction between the charged particles. We obtain the “bright” singlet and triplet trion binding energies and we found that the negatively charged exciton is always less stable than the positively charged exciton in a wire with a hole to electron mass ratio σ>1. The pair correlation functions and the conditional probabilities are calculated, which visualizes the correlation between the particles in the wire.

We introduce a generalized non-equilibrium statistical operator (NSO) to study a current-carrying system. The NSO is used to derive a set of quantum kinetic equations based on quantum mechanical balance equations. The quantum kinetic equations are solved self-consistently together with Poisson’s equation to solve a general transport problem. We show that these kinetic equations can be used to rederive the Landauer formula for the conductance of a quantum point contact, without any reference to reservoirs at different chemical potentials. Instead, energy dissipation is taken into account explicitly through the electron-phonon interaction. We find that both elastic and inelastic scattering are necessary to obtain the Landauer conductance.