Nanoelectronics

The following sections are included:

• Dedication
• Preface
• Contents

The following sections are included:

• The I–V menagerie
• Ohm's law redux
• Fundamentals of quantum transport

The three terminal switch known as the transistor is one of the great inventions and arguably the greatest economic success story of the past century. In the last several decades, commodity prices typically increased by a factor between 3-10X. The cost of a house in the US soared from a mere $20K in the 70s to an average of $200K around 2008. In contrast, semiconductor unit prices have decreased steadily by 6 orders of magnitude! This was made possible by continued improvement in transistor design, but more strikingly because of our ability to integrate many such transistors (currently billions) onto a small chip. To our knowledge, such an economic boom is unparalleled in the history of human existence. If the late 18th and early 19th centuries were characterized by the Industrial Revolution, then the mid to late 20th century was definitely characterized by the rise of digital information technology, with the transistor being its main fuel…

In this chapter, we will recap quantum mechanics and its classical correspondence, and introduce the compact bra-ket notation. Quantum processes clearly underlie nano-electronics, sometimes implicitly through bandgaps and band effective masses, but often explicitly as in tunneling, energy quantization and interference. First quantization, which occupies much of standard texts in quantum mechanics, deals with the quantization of the electron wavelength through suitable boundary conditions on the electron wavefunction , for instance in a box or an atom. The wavefunction satisfies the nonrelativistic Schrödinger equation

While the outcome of a single quantum measurement is inherently uncertain, the odds of measuring a particular outcome are given by the probability density |ψ|² according to the above deterministic equation. That identical electrons can act independently, yet somehow build up collectively to an overall probability distribution, opens up a lot of thorny philosophical quandaries that make quantum mechanics so counter-intuitive. What makes it relevant to technology however, is the fact that most instrumental readings average over many single shot measurements, so we need their aggregate outcomes that can be predicted fairly accurately. We can extract the measured spectrum (eigenvalues) as well as expectation values for various observables by operating on ψ with different Hermitian operators, such as the position operator or the momentum operator , satisfying a commutation relation . Properties such as the Heisenberg uncertainty principle σ_xσ_px ≥ ħ/2 follow naturally from the underlying wave mathematics and commutation algebra, as we discuss shortly (around Eq. (3.5)).

The following sections are included:

• Bonds, bands and resonances
• The hydrogen atom
• Beyond hydrogen: electron-electron interactions
• Role of direct Coulomb repulsion: screening
• Atom tomolecule
• Bonding in molecular ion
• Bond mixing and hybridization

As we extend a set of bonds to form a periodic network, a different symmetry emerges among the eigenstates. The periodicity of the atomic cores in a solid naturally introduces a Fourier space description with an associated wavevector . The allowed eigenstates can now be labeled in terms of , as well as a separate band index at each representing the orbital symmetries of the underlying atomic valence electrons. Near the high symmetry points these E − ks typically look parabolic, and can be interpreted as ‘free’ electrons, albeit with an effective mass m^*. (typically a tensor representing anisotropy). Generally these masses are smaller than the free electron mass, because a valence electron gets pulled in by the ionic cores and slingshots quickly from one atomic well to another in the process…

Let us now discuss how to go beyond single electron quantum mechanics to that of multiple electrons. We will discuss how to account for Pauli exclusion principle properly in the bandstructure, and not just in the occupation statistics. To explore many particle rules systematically, we will need to understand second quantization. Readers may choose to skip this chapter and come back to it later when we discuss interacting systems, particularly, scattering and correlation effects. We will introduce creation and annihilation operators as a way to enforce wavefunction symmetry under particle exchange, and how it leads to the second quantized Hamiltonian. A simple limit is the Hartree Fock theory that marries Coulomb interactions with Pauli exclusion. In this context, we discuss Feynman diagrams as a way to visually describe the underlying processes. We also discuss symmetry of phonon operators, the use of Green's functions to solve their responses to external perturbations, and the concept of self energies as a way to capture complicated interactions not directly included in the electronic potential.

The following sections are included:

• How interactionsmess things up
  • Quadratic electron Hamiltonian: tight-binding
  • Quadratic lattice Hamiltonian: phonons
  • From many body to quadratic: BCS superconductivity
• Green's functions: why we need them
  • Adding interactions: self-energy and Dyson equation
• Appendix: Feynman rules

The aim of this and the following chapters is to discuss some of the core physics behind transport equations that we will use through the rest of the book. We discuss how irreversibility and damping arise out of the evolution equations, and how entropic forces mix with dynamics to generate classical drift diffusion and the Boltzmann equation. Drift diffusion with recombination-generation leads to the minority carrier diffusion equation traditionally used to discuss classical switches and transistors. We then introduce ballistic quantum transport theory, where the entropic terms are conveniently hidden inside the contacts. We derive the nonequilibrium Green's function (NEGF) formalism using simple one electron theory, but present later the many electron counterpart that is especially relevant for strong interactions…

The following sections are included:

• Classical physics and drift-diffusion
• Deriving DD: time-average to ensemble average
• Invoking thermal scattering and white noise
• Fokker-Planck to drift-diffusion

The following sections are included:

• Deriving Boltzmann
• Quasi-Fermi level vs ε-field: what drives a current?
• Recombination-generation: restoring balance
• Minority carrier diffusion equation
• Semiclassical physics: towards Schrödinger

In this section, we will look at the quantum flow of electrons, following a procedure similar to how we derived the classical drift diffusion equation. The latter came from the open boundary Newton's equation, ensemble averaged to describe collective flow. A very similar pathway will now lead to the Keldysh-Kadanoff-Baym Non Equilibrium Green's Function (NEGF) formalism. For simple noninteracting systems, this will simplify significantly, yielding the two terminal Landauer or the multiterminal Landauer Büttiker formalism…

In this and subsequent chapters, we simplify the NEGF equations to show their implications in various limits. In the coherent ballistic limit in absence of interactions, we get the Landauer theory that interprets conductance as the transmission at the Fermi energy. This is in contrast to the Kubo formula operative at low bias, where we interpret conductance as current noise. The Landauer viewpoint leads to quantization of electric and thermal conductance, and interference effects such as Fano oscillations. In dirty samples, the emergence of scattering within the channel leads to Ohm's law and the Drude theory for conduction. In presence of scattering, we also recover the Marcus theory for chemical reactions, Variable Range Hopping for amorphous semiconductors, and Frenkel Poole for molecular solids.

It is well recognized that the Landauer equation in its original form describes the coherent propagation of quantum waves. What is not as well appreciated is how it can also describe classical particulate transport. The relation between the two limits arises, rather elegantly, by the multiple scattering contributions to the mode averaged transmission . Indeed, we can play with the shape of the transmission function to capture all sorts of scattering processes and various ways of averaging over them. We thus have a unified set of equations that work from quantal to classical limits…

In the eventful journey of the electron across molecules and solids, tunneling bears a distinct role as an oft perceived quantum effect. Most quantum phenomena, like interference for instance, require extreme delicacy, typically cryogenic setups so not to perturb the electron's phase. Tunneling however pervades nanoscale systems even under ambient conditions. In fact, standby power dissipation from room temperature oxide tunneling is a major limiter in transistor scaling. While many effects of quantum origin such as bandgaps, magnetism and effective mass can and often are included ex post facto within otherwise classical descriptions, tunneling warrants a thoroughly quantum treatment. The preponderance of tunneling in today's nanodevices makes our quantum kinetic description of current flow all the more relevant to present day industry. In this chapter, we will discuss quantum tunneling and the tunnel current across thin oxide films and molecular wires. In the next chapter, we will deal with the added complexity that arises when tunneling is further symmetry limited by a topological index. We discuss three examples where the molecular nature of the tunneling states matters. These include tunnel magnetoresistance in magnetic tunnel junctions, symmetry filtering across Fe/MgO interfaces, and finally the rich metamaterial behavior of electron waveguiding across graphene PN junctions.

Our treatment of tunneling in the previous chapter and in standard quantum mechanics textbooks requires just a continuum description of the barrier in terms of an overall effective mass. The underlying molecular nature and the ‘bottom-up’ viewpoint may seem like an overkill. However, there are notable examples where the orbital structure of the tunneling electrons imposes additional symmetry rules that constrain the transmission severely. We present three examples in this chapter. The tunnel magnetoresistance across a magnetic tunnel junction is governed by the alignment of majority spins across the oxide. This magnetoresistance can be further tweaked by aligning the orbital structures with the spin structures, so that certain orbital derived bands promote large spin-dependent tunneling and reduce the read current in magnetic memory devices. Finally, 2-D materials like graphene and topological insulators have a band index that we associate with pseudospin in the former and spin in the latter. The alignment of these band indices across a tunnel PN junction makes their transmission strongly angle-dependent and gate tunable, giving us an extra knob to control their device properties.

The key element in designing the conductance of a material is the alignment of its energy bands relative to the contact Fermi energies. The Landauer theory gives us a convenient bottom up way to describe this current, and can be readily modified to include scattering effects (Chapter 10). The Landauer conductance is proportional to the number of modes lying within the conduction window opened up by the applied source to drain bias, times the average transmission probability of these modes. The physics is often dominated by the mode-averaged transmission function alone, such as a WKB function T ~ e^−κL for quantum mechanical tunneling through a bandgap (Eq. (14.3)), an Ohmic contribution for diffusive transport T ~ λ/(λ + L) in a dirty material (Eq. (22.5)), or a sinusoidal term T ~ 1/(Asin² κL + B) for a coherent interferometer (see Fano interference earlier and Eq. (14.3)). We will see later how all these transmissions hail from a common origin (the sin² κL function analytically continues into a decaying exponential sinh² κL outside a band, and phase averages to the Ohmic limit ~ 1/L inside a band). However, a lot of physics, especially with high quality materials and interfaces, arises from bandstructure alone, and their biggest contribution is to the mode density M(E). In what follows, we will focus on transport where the mode spectrum is the key player, in other words, ballistic (T = 1, no velocity scattering) resonant conduction through the middle of a band…

The following sections are included:

• Where are the ‘levels’?
• Where is the Fermi level?
  • A case study: Benzene Dithiol (BDT) on gold
• Molecular rectification
• Molecular dipoles on semiconductors

The multibillion dollar semiconductor industry relies on our ability to significantly alter the current flowing between two terminals, by the application of a voltage to a third, gate terminal. What has made this switching action evolve into a formidable industry is our ability to dope transistors with well controlled doses of trace impurities to make them electron or hole rich, and then to integrate multiple (currently billions) such complementary transistors onto a chip. In complementary metal oxide semiconductor (CMOS) technology, the central unit is an inverter consisting of an n doped and a p doped transistor placed in series between a voltage source and a ground. A common input signal is placed on their gates, so that when one (say the n FET) is turned on by pulling its Fermi energy closer to the majority band, its complement is turned off. This means that the output signal drawn from a point between the transistors shorts through the active FET to either the power supply or the ground, floating thereby to a value that is inverse to the input. What has made this inverter particularly attractive is that due to the serial placement of the complementary FETs, aside from the transient phase when the voltages are changing, the system stays perpetually in open circuit between supply and ground (as either the n or the p MOS is off), and the energy budget is thus quite manageable…

Much of the gating we encounter in device physics is electrostatic, associated with depleting or accumulating charges near the gate terminal at the surface of the channel. Such a process is fast, because of the low effective mass of the electrons. However, it makes it hard to fully turn off the channel, especially when the oxide is very thin, leading to considerable leakage and stand-by power dissipation. Cantilevers and relays allow us to physically turn off the channel by moving pieces out of the conduction path. These processes are relatively slower, but clearly consume considerably less power. Such a gated geometry could vary from an entire cantilever to a fluctuating bond, a rotating molecular ring or a relay protein in a gated sodium channel…

Let us move now from gated structures to purely two-terminal switching phenomena in molecular devices. The terminology ‘switching’ is used loosely, in that we will cover (a) reproducible switching without memory (where the curve is perfectly retraced upon a reverse swing), (b) reproducible switching with memory (i.e., a hysteretic curve), and (c) stochastic switching such as due to occupation/deoccupation of trap states. The detailed underlying mechanisms behind specific switching systems are still under debate, and the absence of systematic experimental data exploring various facets of the same molecule in the same device set-up makes it hard to validate such a model. However, our intent is to discuss the general physics behind such phenomena, and leave it to active researchers to figure out what specific sources of switching are operative in a specific example…

Magnetic fields inject additional dynamics onto electrons on their journey, tightening their rotational orbits and inducing them to cancel each other except at boundaries, where they skip merrily along the perimeters. The channel density of states collapses onto a ladder of equally spaced Landau levels representing bound states in the magnetic oscillator potentials. The levels are broadened by interactions with impurities and contacts (i.e., a self-energy), and occupied by electrons upto a Fermi energy. The placement of the Fermi energy on or off resonance with the Landau levels leads to dips and plateaus in the Hall voltage and show quantization effects. On the other hand, the interaction between the Landau levels and a commensurate background lattice potential yields a quasi-periodic phase plot known as the Hofstadter butterfly. Finally, collections of spins, exchange coupled to form a magnet, can impose their spin polarization onto electron currents passing through. When these polarized spins impinge on a soft magnet misaligned with them, angular momentum is exchanged between the two groups. The torque flips the magnet and allows us to write information onto it as a scalable, non-volatile memory. Once again, our dynamical equations allow us to explore these effects readily.

Along the electron's journey, it is likely to encounter multiple scattering events, potholes and road blocks, bridges and bus stops, that hinder, scramble and sometimes even accelerate its flow. The most familiar are momentum scattering events that create an electron backflow at an impurity site and generate a net resistance. As we saw earlier, a contact resistance appears even for a ballistic channel, as electrons from numerous contact modes M struggle to coalesce into a few channel modes m. The contact resistance, curiously, is a universal constant R = h/2q² for a single moded double spin channel, because the current is set by (Eq. (12.5)), but the transmission probability itself goes as for m = 1 ≪ M, as we expect near a good contact. The resistance is akin to the buildup of a large incoming traffic merging from a six lane highway to a single lane. The opposite process creates no bottleneck and thus no resistance for electron escape into the contacts…

The following sections are included:

• Boltzmann and irreversibility
• Does scattering help or hurt?
• A case study: minimum conductivity in graphene in presence of charge puddles
• A variety of scattering mechanisms

The following sections are included:

• Scattering in NEGF: the self-consistent Born approximation (SCBA)
• Beyond SCBA: higher order terms
• The information content in the scattering matrix
• When scattering helps: non-resonant Inelastic Electron Tunneling Spectroscopy (IETS)
• Near resonant vibronic signatures — polarons and sidebands
• Current saturation fromphonon emission
• Comparison with photon assisted tunneling
• Can phonons be coherent?

In most studies on electronic devices, we spend little time on interactions among charges, beyond just an average potential that each electron swims in and contributes to at the same time. Complexities due to spin and Pauli exclusion are usually included ‘after the fact’. We take care to avoid double occupancy but do not worry about their impact on the bandstructures themselves. We worry about spins when they can be separately manipulated with magnetic fields, strain, spin-orbit coupling or exchange. What is less prevalent in device models is any distinction between spins in a purely electrostatic environment. But Pauli exclusion forces such a distinction and creates internal magnetic fields that correlate the charges and spins, especially when they need to avoid each other in the presence of Coulomb repulsion. In 3d and 4f transition metals with localized electrons (l = n − 1, so no radial nodes), quantum dot arrays and molecular chains with large unit cells, the overlap of electrons setting their kinetic energies is small compared to their Coulomb cost. A rich tapestry of physical phenomena arises from such strong correlation effects, effects that dominate whenever the energy cost of adding an electron to a level exceeds its broadening by the environment. They range from implicit effects such as many body corrections to band-gaps or optical spectra, to more explicit effects such as superconductivity, superfluidity, polarons, ferromagnetism, excitons and heavy fermions, where particles devise ingenious ways to coordinate their dynamics in order to cut down their overall energy cost…

The following sections are included:

• Equilibrium transition rates between states
• Nonequilibrium transitions between many body states
  • Role of excitations
  • The ‘master equation’
  • Deriving the master equation
  • Lumping excitations together: the orthodox model

In Fig. 25.1 we showed the equilibrium ground state occupancy as we ramped up the Fermi energy. The red dashed line came from adding a self consistent mean field obtained by Taylor expanding the Hamiltonian from Eq. (25.8) around the mean and ignoring products of small fluctuations, known as the Hartree Fock or self consistent field approximation

The following sections are included:

• Looking back: a microscopic view of transport physics
• Looking ahead: the high cost of switching
• Limit to dissipation: the Boltzmann ‘tyranny’
• Bypassing the ‘Boltzmann tyranny’ — quantum physics to the rescue?

Let us put our theoretical machinery to the test — by exploring the subthermal switching of transistors beyond the Boltzmann limit. As discussed in the previous chapter, we require a gate dependent change in shape of the transmission function, in addition to the shift in Fermi energy relative to the band-edge, to draw more current than usual. We will discuss a few ways to do this. Many of these devices may prove inadequate in retrospect; — they may have other trade-offs, or end up too expensive, unreliable or difficult to integrate into existing CMOS process flows. The primary purpose of this exercise is to showcase how we can take a practical problem and walk through an analysis using the Landauer-NEGF approach.

The following sections are included:

• Biology's secrets
• Electronics with molecules?

The following sections are included:

• References
• Index