Quantum theory is cast in the language of complex vector spaces. These are mathematical structures that are based on complex numbers. We learned all that we need about such numbers in Chapter 1. Armed with this knowledge, we can now tackle complex vector spaces themselves.

Section 2.1 goes through the main example of a (finite-dimensional) complex vector space at tutorial pace. Section 2.2 provides formal definitions, basic properties, and more examples. Each of Section 2.3 through Section 2.7 discusses an advanced topic.

I am now ready to move on to the transport theories based on Viewpoint 2: the field is a consequence of carrier flow (see Sec. 1.6). Seeds of this approach are almost as old as quantum mechanics. For instance, one of the first applications of scattering theory to determine the electrical resistance at the contact between two conductors can be found in the work of J. Frenkel published in 1930 (Frenkel, 1930). Here, the calculation of the current is done by assuming that electrons on the two sides of the contact, at their own local equilibrium distribution, have a finite transmission probability to tunnel across the potential barrier induced by the junction. This result anticipated the concept that, under specific conditions, the conductance of a given system sandwiched between electrodes can be related to its transmission properties, namely to the probability for electrons to “cross” the system in going from one electrode to the other.

However, the idea that scatterers can induce, self-consistently, local fields which “act back” on the carrier dynamics was pioneered by Landauer (Landauer, 1957), and has contributed tremendously to our understanding of electron transport in mesoscopic and nanoscopic systems. It is a major conceptual departure from the theories I have discussed in the previous chapter.

Another fundamental result of the Landauer approach is that a finite resistance emerges even if the transmission probability of the sample is unity; a fact I have anticipated – though from a different perspective – in Sec. 2.3.4. This result requires a non-trivial understanding of the role of the sample versus all other elements of an electrical circuit (Imry, 1986).

So far I have discussed the most obvious quantity in a transport problem: the average current. In reality, as I have anticipated in Sec. 1.2, electrical current continually fluctuates in time so that it carries noise.

There are several sources of noise in a conductor that can be classified as external and internal. Two of the most common sources of external noise are known as 1/f and telegraph noise. The first takes its name from the fact that it shows a spectrum at small frequencies f, which scales as 1/f. No general theory exists on this noise, and it is believed to be of extrinsic origin, for instance due to distributions of defects in the conductor. The interested reader should consult the review by Dutta and Horn (1981) where 1/f noise is discussed at length.

On the other hand, telegraph noise is due to rapid fluctuations between states of the conductor that are very close in energy, but that carry different currents. This noise is also believed to be of extrinsic nature and has a typical Lorentzian power spectrum.

In addition to the above fluctuations, one has to remember that our nanoscale system (sample), whose noise properties we want to determine, is connected to a real external circuit with its own external resistance, call it Rext (Fig. 5.1). Therefore, due to the presence of this extra resistance, even if the voltage source is noiseless (call the corresponding potential drop Vsource), the voltage drop across the sample (call it Vsample) is different from Vsource and would manifest fluctuations. This is easy to see using the following simple arguments.

So far, we have dealt mostly with the electron system already at an ideal steady state. We never questioned whether this state exists at all, and if so, how a many-body system does actually reach that steady state, whether the steady state we impose via single-particle scattering boundary conditions is actually what the electrons want to realize when they flow across a nanojunction, or even if it is unique.

In addition, we have mostly worked with electrons interacting at a mean- field level (see discussion in Sec. 4.2.4). We discussed in Chapter 4 how, in principle, one can introduce electron-electron interactions beyond mean field using the non-equilibrium Green's function formalism. However, except for simple model systems, it is computationally demanding – and most of the time, outright impossible – to use the interacting version of the NEGF practically in transport calculations (and not only in transport).

A simpler and more efficient way to treat electron-electron interactions in a transport problem would be thus desirable.

In this chapter I will introduce an alternative picture of transport – I will name it micro-canonical (Di Ventra and Todorov, 2004) – that does not rely on the approximations of the Landauer approach. In addition, this formulation does not require partitioning the system into leads and a central region, or assuming the leads contain non-interacting electrons in order to have a closed form for the current.

In fact, within this picture I will prove several theorems of dynamical density-functional theory (Appendices E, F and G) that, in principle, allow us to calculate the exact current – namely with all many-body effects included – within an effective single-particle picture.

Nanoscale systems

Let us briefly discuss the systems I will consider in this book, those of nanoscale dimensions (1 nm = 10–9 m). The phenomena and theoretical approaches I will present are particularly relevant for these structures rather than those with much larger dimensions.

So, what is a nanoscale system? The simplest – and most natural – answer is that it is a structure with at least one dimension at the nanoscale, meaning that such dimension is anywhere in between a few tens of nanometers and the size of an atom (Di Ventra et al., 2004a). One can then define structures with larger – but still not yet macroscopic – dimensions as mesoscopic. This separation of scales is arguably fuzzy. Mesoscopic structures share some of the transport properties of nanostructures; the theoretical description of both classes of systems is often similar; and in certain literature no distinction between them is indeed made.

Is there then, in the context of electrical conduction, another key quantity that characterizes nanoscale systems? As I will emphasize several times in this book, this key quantity is the current density – current per unit area – they can carry. This can be extremely large.

As an example, consider a wire made using a mechanically controllable break junction (Muller et al., 1992), a junction that is created by mechanically breaking a metal wire. Such a structure – a type of metallic quantum point contact – may result in a single atom in between two large chunks of the same material (see schematic in Fig. 1.1).

Within the Landauer approach to conduction discussed in the previous chapter, electron interactions have been included only at the mean-field level. This is quite a strong approximation, especially in nanojunctions, where large current densities are common. I therefore want to go beyond this level of description.

I can follow two different routes. (1) I can develop a functional theory of quantum correlations via time-dependent effective single-particle equations, as I will do in Chapter 7 where I will use dynamical density-functional theories within the micro-canonical approach to conduction. (2) I can try to solve the time-dependent Schrödinger equation 1.16 – or its mixed-state version, the Liouville-von Neumann equation 1.60 – directly.

Written this way, this last proposition seems hopeless. In reality, we can employ a many-body technique, known as the non-equilibrium Green's function formalism (NEGF), also referred to as the Keldysh formalism (Keldysh, 1964; Kadanoff and Baym, 1962), which allows us, at least in principle, to do just that: solve the time-dependent Schrödinger equation for an interacting many-body system exactly, from which one can, in principle, calculate the time-dependent current. This is done by solving equations of motion for specific time-dependent single-particle Green's functions, from which the physical properties of interest, such as the charge and current densities, can be obtained.

I have stressed the term “time-dependent” several times, because, as I will show in a moment, the NEGF is “exact” only when one solves the time-dependent Schrödinger equation for a closed quantum system, subject to deterministic perturbations: the system is closed but not necessarily isolated. These perturbations may drive the system far away from its initial state of thermodynamic equilibrium (Keldysh, 1964).

Electrical current is affected by the interaction between electrons and ions. Due to this interaction electrons may undergo inelastic transitions between states of different energy, even if the electrons themselves are considered non-interacting with each other. These transitions appear as discontinuities (steps) in the current (conductance) at biases corresponding to the phonon spectrum of the structure. In reality, the phonon spectrum is renormalized by both the electron-phonon interaction at equilibrium, and by the current itself. The latter fact makes the concept of phonons under current flow fundamentally less obvious. I will discuss this point in Sec. 6.5.

An example of inelastic features in nanoscale systems is illustrated in Fig. 6.1 where the conductance of a gold point contact is measured as a function of bias. The conductance shows a step in the range between 10 and 20 meV corresponding to the energy of the vibrational modes of the whole system – gold point contact plus electrodes – that couple more effectively with electrons.

Via the same inelastic mechanism, electrons can exchange energy with the ions and thus heat up the nanostructure while they propagate across it. As we will see later, this phenomenon, called local ionic heating, may have dramatic effects on the stability of nanostructures.

Finally, in a current-carrying system ions may be displaced by local current-induced rearrangements of the electronic distribution – the local resistivity dipoles I discussed in Sec. 3.2 – without the intervention of inelastic processes. The forces responsible for such displacements are known as current-induced forces. Despite many studies, past and present, these forces challenge our understanding of non-equilibrium phenomena, starting from their basic definition for a current-carrying system to their, yet unsolved, conservative character.

In this appendix I outline the main tenets of time-dependent current DFT (TD-CDFT). A pedagogical and extended account of this approach can be found in the book by Giuliani and Vignale (2005).

The current density as the main variable

In Appendix E I have hinted at the problem of obtaining a local representation of the exchange-correlation scalar potential Vxc (the one appearing in the Kohn–Sham equations E.4) in terms of the density. Without going into details, I just mention that the physical reason why such a representation does not exist for an inhomogeneous electron liquid is related to the strong non-local (in space) functional dependence of this potential on the density.

A way out of this problem is to realize that due to gauge invariance a time-dependent scalar potential V(r,t) can always be represented by a longitudinal vector potential A(r,t).1 We also know that the density is related to the current density via the continuity equation 1.11. Putting these two facts together, we therefore see that the response of the density operator to an external scalar potential can be expressed as the response of the current density operator to an external vector potential, i.e., we can reformulate time-dependent density-functional theory in terms of the current density instead of the density. It turns out that a local representation of the exchange-correlation vector potential in terms of the current density can be derived.

Let us consider a closed system whose degrees of freedom can be divided into two distinguishable sets – call them S and B, e.g., electrons and phonons – and we are interested in the dynamics of only one of the two, say, the electrons. Call the set of degrees of freedom S the “system”. These two sets of degrees of freedom are mutually interacting, but do not exchange particles, namely the number of particles of S is fixed.

Let us also suppose that the other set of degrees of freedom B is so large that we are not interested in its microscopic dynamics, or it is simply impossible to calculate. For both mathematical and physical reasons, by “large” I mean infinitely large. This set of degrees of freedom then acts as an environment for the system S (Sec. 1.2).

Given an initial condition, the dynamics of S + B is reversible, so that it is generated by a group of unitary operators U (t) (Eq. 1.18) on the Hilbert space of S + B, that depends on one parameter: the time t. On the other hand, if we follow only the degrees of freedom of S, by considering the environment B infinite (hence with an infinite Poincaré recurrence time – Sec. 1.2.1), we impose a preferential direction of time because, due to the interaction of S with B, during time evolution some correlations in the system S are “lost” into the degrees of freedom of B without the possibility to recover them (see Sec. 2.8).