Nanoelectronics

A Molecular View

Avik Ghosh

• Publisher: WSPC
• Year: 2016
• ISBN:
  9789813144514
• Topic: Physical Sciences
• Subtopic: Nanoscience

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This book is aimed at senior undergraduates, graduate students and researchers interested in quantitative understanding and modeling of nanomaterial and device physics. With the rapid slow-down of semiconductor scaling that drove

PART 1

Nano-Electronics:

Cool Science vs Smart Technology

Chapter 1

The intellectual lure — quantum physics in devices

1.1 The I–V menagerie

We begin with the physical motivation, primarily to explore the dynamics of electrons as they journey along a channel. Along the way, electrons can encounter a rich variety of interactions affecting their dynamics, at times acting as classical point particles ricocheting off each other, alternately as quantum waves interfering and even annihilating each other. Electrons can lock their spins in a magnet or correlate their charges in a condensate, in a delicate dance aimed to reduce their overall energy cost, mediated by quasiparticles such as collective atomic vibrations in a solid.

To do justice to such a wide range of properties especially far from equilibrium, we must start at the bottom. We need to develop the dynamical equations governing electron flow, including its environment at its most fundamental, molecular level. We need to do this in a unified way, spanning organic molecules, solids, magnets, and the continuum of materials in between. The aim of this book is to provide such a ‘molecular’ view of nanoelectronics, and see how this may help us understand or even design novel nano-electronic devices. This is not just to say that we develop the transport equations ‘bottom-up’. We go further by borrowing tools from molecular chemistry, such as orbital basis sets relevant for benzene or Hartree Fock descriptions of exchange, moving up to a large piece of solid, touching systems in between such as two dimensional materials and strongly correlated quantum dots.

What is the merit of such a detailed molecular deconstruction? Many simulators for solid state devices still rely on classical, Newtonian drift diffusion equation, with quantum mechanical effects introduced as parametrized corrections (one may even designate some of them as ‘fudge factors’). This worked in the past, even if it required a large proliferation of unphysical parameters in compact models. The situation, however, has changed radically over the last several years. Commercial transistors as of 2014 are already operating at more than 70% of their ballistic limit. This means that the observed current is 0.7 times what it would have been if the electrons went ballistically, i.e., single shot like a bullet with no momentum back scattering. Channels studied in emerging research such as carbon nanotubes or graphene are often measured to be nearly 100% ballistic. The few straggling, diffusing electrons still keep transistors operating fairly classically, but to properly account for ballisticity in emerging devices, we will increasingly need to adopt a ‘bottom-up’ view of resistance, given that there is seemingly nothing to resist the bullet-like flow of a ballistic electron except the contacts.

There are other quantum effects, notably tunneling — the tail of an electron wave zipping right through a thin barrier — that are mainstream today. Electron tunneling through oxides into the gate drove up the standby power dissipation and was a significant problem in pre high-k insulator technology days. Direct source to drain tunneling still continues to plague ultrashort materials such as organic molecules on metal contacts. At the same time gate controlled tunneling creates new opportunities for switching devices such as resonant quantum dots and subthermal tunnel field effect transistors (TFET). Equally interesting are strong many-body correlations, which are especially challenging to model far from equilibrium, but highly relevant to emerging device concepts such as Mott switches. These effects are hard to describe even qualitatively using bulk descriptions like effective mass and mean-field potentials. All this is not surprising in retrospect, given that one can arrive at classical dynamics as a limiting case for a quantum particle, by adding in dephasing events in bulk materials. However, there is no honest way to add stuff to classical physics, and hope to emerge naturally with quantum attributes in the end. The same applies for many body correlations, especially at non-equilibrium; they do not come out naturally as limiting cases of conventional ‘mean-field’ device models where electrons interact with other electrons through average potentials.

The emphasis in this book is on a molecular view of the flow of current far from low bias, where strong nonequilibrium physics persists. Far from equilibrium, a fascinating variety of current voltage (I-V) characteristics can be observed, from linear to exponential to saturating to asymmetric to a sequence of cascaded plateaus (Figs. 1.1-1.3). Much of biology and chemistry involves reaction kinetics and phase transitions that are meaningful away from equilibrium. But even to a device engineer, each such I-V curve bears potential significance in the nonlinear regime — as a tunnel device, rectifier, transistor, modulator or single electron switch for instance. It is the aim of this book to bring such a wide variety of transport phenomena under a common umbrella, especially with strong correlation and incoherent scattering, evolving from a unifying set of principles and equations. We can use these principles to explore new emergent materials (say a topological insulator), or new physical principles (e.g., spintronics). At the same time, they should connect with familiar devices such as a regular CMOS transistor or a pn junction. Thus the aim of this book is not just to study the emerging ‘bottom-up’ physics, but also the conventional ‘top-down’, and then see how they meet up in the middle.

1.2 Ohm’s law redux

The quantum vs classical distinction manifests itself not just in exotic devices, but in mainstream systems as well. We can trace this by looking at Ohm’s law, which is taught in every physics and microelectronics curriculum. The classical resistance of a device is given by the voltage V to current I ratio, R = V/I = L/σA, where A is the cross sectional area, L is the length and σ is the conductivity of electron flow. Loosely speaking, the electron flows like fluid in a hose pipe or perhaps grains of sand in a sieve. The fluid resistance is high if we make the bore of the pipe narrow and reduce A, make L very long so the fluid scatters more often from inclusions or pieces of dirt, or increase the ‘viscosity’ of the fluid by replacing water with tar and thus reduce the conductivity σ. The equation was derived empirically, made intuitive sense and was measured repeatedly over the years.

The new Ohm’s law, which has been operational since the 80s, arose when the channel sizes shrunk to the point that the electrons shot across ballistically like a bullet and the average scattering length exceeded the channel length. Since the channel offers little resistance as a result, the resistance of the electron must be mandated by the contacts. Surprisingly however, the ultimate low bias ballistic resistance for a thin, single moded conductor is independent of the contact properties. In fact, the fundamental resistance ends up as a universal constant R₀ = h/2q² ≈ 12.9 kΩ, where electron charge q = 1.602 × 10⁻¹⁹ Coulombs and h = 6.602 × 10⁻³⁴ Js. The presence of Planck’s constant h suggests that the origin is quantum mechanical.

Fig. 1.1 The I-V menagerie showing classical I-Vs (top row) vs quantum systems with tunneling and interference (bottom row). All are described with Landauer theory with conductance set by the quantum transmission probability summed over modes (Eq. (1.1)). Parts 3-5 of the book deal with ballistic quantum transport.

Fig. 1.2 The I-V menagerie, showcasing examples of scattering. The treatment of inelastic scattering (bottom row) requires us to invoke the full machinery of Nonequilibrium Green’s functions (NEGF) beyond just the Landauer transmission formalism. Part 6 of the book focuses on the impact of scattering

Fig. 1.3 The I-V menagerie, showcasing examples of correlated transport in strongly interacting systems that require us to go beyond ‘mean field theory’. We can get these I-Vs by working directly with the density matrix in many body configuration space (Part 7 of the book). We could revert to conventional NEGF if we can extract an approximate self energy, typically nonlocal in time and non-perturbative in the interaction parameter.

We will have occasion to discuss this at length in this book, as well as the behavior of the resistance for multimoded conductors far from low bias. The reconciliation of the two limits, classical vs quantum, arises from a restatement of Ohm’s law as

where M is the number of modes in the channel, and

is the average quantum mechanical transmission probability (ratio of transmitted to incident current densities), a number lying between zero and one. We will find that this resistance can be separated out into

so that the quantum and classical contribution...

Table of contents

• Cover page
• Title page
• Copyright
• Dedication
• Preface
• Contents
• Part 1 Nano-Electronics: Cool Science vs Smart Technology
• Part 2 Equilibrium Tools: Quantum Mechanics and Thermodynamics
• Part 3 Nonequilibrium Concepts: The Nature of Transport Equations
• Part 4 Adventures and Applications: Ballistic Quantum Flow
• Part 5 Controlling Electron Flow: Switching and Gating
• Part 6 A Smattering of Scattering: Resistance, Decoherence and Dissipation
• Part 7 When Electrons Tango! Dealing with Nonequilibrium Correlations
• Part 8 Looking Back and Looking Ahead: Novel Transport vs Novel Devices
• References
• Index