Metal-oxide semiconductor field-effect transistor scaling is following the prediction of the Moore's law enunciated in 1965 [5]. So far, this trend for miniaturization has never been invalided, enabling the industry of semiconductors to cope with the everlasting demand for higher performance at lower cost. However, this scaling becomes increasingly difficult to follow due to inherent process and device performance limitations for technology nodes beyond tents of nm. To stand the pace of downscaling, nonclassical device architectures have been continuously proposed in the ITRS roadmap.

The junctionless field-effect transistor is one of these nanoelectronics devices that is expected to withstand the downscaling of Complementary Metal-Oxide– Semiconductor (CMOS) technology by enabling easier fabrication processes, while allowing high performance. In addition, semiconductor nanowires largely used for label-free biosensing are essentially junctionless FETs without any gate.

Therefore, since the first implementation of junctionless FET nanowires by J. P. Colinge in 2009, growing interest in these devices in different fields of research motivated the authors to write a book dedicated to analytical modeling of double-gate and nanowire junctionless FETs. In contrast to the abundant literature on modeling and compact modeling of inversion-mode MOSFETs, analytical modeling of field-effect transistors without junctions is still following different strategies.

After discussing the advantages and limitations of junctionless field-effect transistors in the first introductory chapter, a thorough overview of published analytical models for double-gate and nanowires configurations is presented in Chapter 2, including the mains assumptions which are introduced.

In Chapter 3 and beyond, the analytical model of the so-called EPFL junctionless field-effect transistor is presented. After discussing the roots ending with a chargebased model valid in all the regions of operation, important features are introduced gradually in Chapters 4–13, each of which targets a specific feature. These topics include nanowire versus double-gate equivalence, technological design-space, junctionless FET performance, short-channel effects, transcapacitances, asymmetric operation, thermal noise, interface traps, and a revisited model for JFETs. In addition a general mobility extraction technique is proposed.

We suggest readers see Chapter 3 for a general introduction to the charge-based model before proceeding further. This is where the main ideas are introduced that will thus be used in the following chapters.

The Birth of the Transistor

Since its invention in 1907 by Lee de Forest, the vacuum tube has been a major driver for electronics and communications technologies. However, vacuum tubes have conceptual limits. Besides being quite expensive, they have reliability issues, high energy consumption, and miniaturization limits. To address these limitations, AT&T, one of the main US phone companies, tasked its R&D unit – Bell Laboratories – with the development of an innovative device that would be cheaper, smaller, and more reliable – in short, a good substitute for the vacuum tube.

The initial idea for “transistors” came from Edgar Julius Lilienfeld in 1925 [6]. The scientist patented in Canada (1925) and successively in the United States (1928) the first field-effect semiconductor device designed to change the resistivity with applied voltages. However, due to the lack of dedicated technology, Lilienfeld never had the chance to validate his concept. It's interesting to note that his transistor idea was quite similar to today's accumulation-mode (AM) transistors.

The point-contact transistor built by John Bardeen and Walter Brattain at Bell Laboratories in the United States in December 1947 was the first solid-state transistor ever created. The two scientists, led by the physicist William Bradford Shockley, discovered the new amplification device while working on theory and experiments on solid-state materials (germanium in this case). This first proof of “controlled resistance” and “amplification” in a solid-state device is considered as the birth of the transistor, and its inventors were jointly awarded the Nobel Prize in physics in 1956 for this achievement.

After invention of the point-contact transistor, Bell Labs followed up with design and fabrication of solid-state amplifiers, gaining a deep understanding of transistors physics. A fully working junction transistor based on Shockley's design, consisting of adjacent semiconductors junctions, was fabricated in 1951 [7]. In 1954, at Texas Instruments Gordon Teal demonstrated the first silicon transistor using the knowledge gained by growing high purity crystals while working at Bell Labs.

The Metal-Oxide–Semiconductor Field-Effect Transistor

By the end of 1950s, all solid-state transistors were bipolar junction transistors (BJT).

Introduction

Monte Carlo methods indicate a broad class of numerical algorithms that are based upon repeated random sampling to obtain the solution of several mathematical and physical problems. The typical issue is about the calculation of large sums or integrals and the revolutionary idea is that we do not perform an exact enumeration/ integration but instead we generate random samples, which are then added together to approximate the exact result. Therefore, the concept of sampling is pivotal in any Monte Carlo approach: its meaning is just to produce random examples of configurations (e.g., N classical particles distributed in a box) that are used to give an accurate estimate of the exact quantity under examination (e.g., the internal energy at fixed temperature).

The roots of the Monte Carlo approaches date back to Enrico Fermi's attempts while studying neutron diffusion in the early thirties. He did not publish anything on the subject, but he got the credits for the idea that a time-independent Schrodinger equation can be interpreted in terms of a system of particles performing a random walk (Metropolis and Ulam, 1949). Few years after, a fundamental step forward has been done by Stanislaw Ulam, when he was working on nuclear-weapon projects at the Los Alamos National Laboratory. His first thoughts about Monte Carlo methods were suggested by a question that occurred when he was convalescing from an illness and playing solitaire. Ulam tried to calculate the likelihood of winning based on the initial layout of the cards. After exhaustive combinatorial calculations, he decided to go for a more practical approach of trying out many different layouts and observing the number of successful games. At that time, a new era of fast computers was beginning and John von Neumann understood the relevance of Ulam's suggestion and proposed a statistical approach to solve the problem of neutron diffusion in a fissionablematerial. Ulam and von Neumann worked together to develop algorithms including importance sampling and rejection sampling, which are at the basis of any modern Monte Carlo technique. In addition, von Neumann developed a way to obtain pseudo-random numbers by using the so-called middle-square digit method, since he realized that using a truly random number was extremely slow.

Introduction

By using the laws of classical mechanics, in principle we can make exact predictions of events by knowing the exact initial conditions (i.e., all positions and velocities of the relevant degrees of freedom). However, in practice, there are several events that are unpredictable, essentially because it is impossible to have the exact knowledge of the initial conditions and a very small error in those conditions will grow exponentially in time, invalidating any attempt to follow the exact equations of motion. When tossing a coin or rolling a die, we do not know the outcome of the event, but we can give some probability to each event, e.g., 1/2 for head and 1/2 for tail when tossing a coin, or 1/6 for each side of a die (we assume that we are playing with fair coins and dice). The probability gives the measure of the likelihood that a given event will occur. Of course, it is based upon a mathematical approach, which transforms the unpredictability into something that is somehow predictable.

This idea seems very simple, but it took several hundred years to capture it. Indeed, since the ancient times, people in Greece and in the Roman Empire (but also in Asia, for example, in India) were tenacious gamblers; nevertheless, nobody tried to understand how the random events were related to mathematical laws.Many quarrels and disputes were resolved by tossing a coin, and the result was seen as the manifestation of the “celestial will.” The human superstition represented a huge obstacle to define a scientific (i.e., mathematical) approach to random events. Eventually, after several hundreds of years, superstition was overcome by an even stronger human impulse: the desire of obtaining an economical profit.

The birth of the mathematical theory of probability is due to the studies done by Girolamo Cardano, who realized that for equiprobable events (like tossing a coin or rolling a die), the probability that a single event will appear is equal to 1 over the number of all possible events (independently from any celestial will).

Quantum Averages and Statistical Samplings

In this chapter, we discuss the general framework in which the variational Monte Carlo methods are defined and few important implementations for interacting systems of bosons and fermions on the lattice. The main advantage of considering this approach relies on the variational principle that has been shown in section 1.4: the energy of a given quantum state is always bounded from below by the exact ground-state one, giving us the route to obtain the best possible solution to the problem. In most cases, a suitable parametrization of the variational wave function allows us to consider a wide range of different quantum phases (e.g., metals, superconductors, and insulators). By performing an optimization of the parameters, we can reach the lowest-energy state, which is expected to capture the correct ground-state behavior. Therefore, variational wave functions represent a flexible and valuable approach to get important insights into the low-energy properties of models that cannot be solved by exact methods. By contrast, the main limitation of this approach is the fact that it is based on a given Ansatz, which may contain a relevant bias that cannot be removed within the chosen parametrization.

We also mention that variational wave functions can be easily defined and treated for a wide class of models, irrespective of the range of interactions and the dimension of the local (i.e., single-site) Hilbert space, which can be even infinite and does not need the use of an uncontrolled cutoff to work with a finite-dimensional space. In this respect, variational Monte Carlo is better than other methods, like densitymatrix renormalization group or tensor-network methods in which the complexity dramatically increases with both the range of the interaction and the dimension of the local Hilbert space (White, 1992; Schollwöck, 2005, 2011).

Let us start by describing the general framework in which variational Monte Carlo methods are defined. First of all, we fix a complete basis set in the Hilbert space, in which (for simplicity) the states are taken to be orthogonal and normalized such that:

Then, any quantum state can be written as:

In turn, the expectation value of an operator over a given variational wave function takes the following form:

The main problem in evaluating the expectation value is that the number of configurations in the sum is exponentially large with the number of particles.