1
Introduction*

"Dimensionality" and "atomic ordering in finite structures" seem like rather odd principles by which to organize thoughts on solid-state physics. Indeed, this is not a historical approach to understanding solids at all. However, in learning solid state today, we must embrace the historical orthodoxy of crystal lattices, phonons, and band structure, as well as a whole zoo of emerging exotic materials that range from fullerenes to organic superconductors.

How do we understand two-dimensional () dichalcogenides, atomically layered permanent magnets, perovskites, topological insulators, conducting polymers, quantum dots, graphene, glassy carbon, etc.? And what of the low-dimensional analogues of orthodox collective behavior: charge density waves, excitons, spin waves, and the like? We know these things "live" in/on such low-dimensional structures. An interesting and instructive way to build a framework is to begin with the normative behavior of a special atom, carbon, and the dimensionality of the structures it makes. Why carbon? Because among the elements it is about the most robust at making compounds and structures. It is extremely flexible in how it chooses to arrange itself. Why dimension? Well, lower-dimensional materials offer new approaches to technology, holding the key to everything from quantum computers to new medicines. But most importantly, it introduces the idea of "topology."

Look, the traditional story goes like this. We begin our description of solids with an infinite mathematical construction (the lattice) given by specific point group symmetries. Onto the lattice points we attach some arbitrary set of atoms (generally picking something found in nature). We calculate specified properties based upon idealizations of how free the electron may be at each lattice point or how free the motion of the atom at the lattice point may be. We adiabatically add interactions between vibrations, carriers, etc. of the lattice to get more interesting phenomena.

Our story, though, is like a tale of die Brüder Grimm1: carbon is the central atom of the universe.2 It forms more compounds in more ways than any other atom. Thus, other atomic systems deviate from carbon by breaking its norms of symmetry. Beginning with large carbon molecules, we form nanometer structures. As we add, subtract, or substitute C atoms in the structure, we design materials with properties that can be examined through the dimensional change we have brought out. It isn't quite a chemical point of view, and it isn't quite solid-state physics in its purest form. It is the type of conversation you hear in working research labs across the world: complementary and an enjoyable compromise between the perspectives.

1.1 Dimensionality

The concept of dimensionality has been with us for a while, and it is an intellectually appealing concept. Speaking of a dimensionality other than three will surely attract some attention. Some years ago it was fashionable to admire physicists who apparently could "think in four dimensions" in striking contrast to Marcuse's One-Dimensional Man (Figure 1.1) [1]. Physicists would then respond with the understatement: "We only think in two dimensions, one of which is always time. The other dimension is the quantity we are interested in, which changes with time. After all, we have to publish our results as two-dimensional figures in journals. Why should we think of something we cannot publish?"

Figure 1.1 Marcuse's man. Simultaneously with Herbert Marcuse's book One-Dimensional Man, which widely influenced the youth movement of the 1960s. W.A. Little's paper on "Possibility of Synthesizing an Organic Superconductor" was published, motivating many physicists and chemists to investigate low-dimensional solids.

This fictitious dialogue implies more than just sophisticated plays on words. If physics is what physicists do, then in most parts of physics there is a profound difference between the dimension of time and other dimensions, and there is a logical basis for this difference [2]. In general, the quantity that changes with time and in which the physicist is interested is some intrinsic property of an object. The object in question is typically imbedded in a three-dimensional () space. Objects themselves, however, may be very flat such as flounders, saucers, or oil films with greater length and width than thickness. In materials such as graphene or MoS2, thickness can be negligibly small - atomic. Such objects can be regarded as (approximately) 2D. Now, if the intrinsic property that the physicist wishes to study is somehow constrained in behavior, in direct correlation to the dimension of the object, like a boat on the 2D surface of the sea is hopefully constrained to 2D motion, then we say the property is expressing the dimensionality of the object. In our everyday experience one-dimensional () and 2D objects and 1D and 2D constraints are more common than you might think. Indeed, low dimensionality should not be particularly spectacular to our expectations. For this reason too, it is reasonable to introduce non-integer, or fractal, dimensions [3]. Not much imagination is necessary to assign a dimensionality between one and two to a network of roads and streets - more than a highway and less than a plane. It is a well-known peculiarity that, for example, the coastline of Scotland has the fractal dimension of 1.33 and the stars in the universe that of 1.23.

Solid-state physics treats solids both as objects and as the space in which objects of physics exist, e.g. various silicon single crystals can be compared with each other, or they can be considered as the space in which electrons or phonons move. The layers of a crystal, like the ab-planes of graphite, can be regarded as 2D objects with interactions between them that extend into the third dimension. But these planes are also the 2D space in which electrons move rather freely. Similar considerations apply to the (quasi) 1D hydrocarbon chains of conducting polymers.

1.2 Approaching Dimensionality from Outside and from Inside

There are two approaches to low-dimensional or quasi-low-dimensional systems in solid-state physics: geometrical shaping as an external approach and increase of anisotropy as an internal approach. These are also sometimes termed top-down and bottom-up approaches, respectively. For the external approach, let us take a wire and draw it until it gets sufficiently thin to be 1D (Figure 1.2). How thin will it have to be to be truly 1D? This depends a little on exactly what property of the structure is desired to express low-dimensional behavior. Certainly, thin compared to some microscopic parameter associated with that property. For example, for 1D electrical transport properties, the structure must have length scales such that the mean free path of an electron or the Fermi wavelength is affected by the physical confinement of the structure. We will discuss these concepts further a little later on in the text. But surely the meaning is clear: some fundamental aspect of an internal object responsible for the phenomenon of interest must be dramatically altered by its localization within the structure.

Figure 1.2 Wire puller. An "external approach" to one-dimensionality. A man tries to draw a wire through a mandrel until it is thin enough to be regarded as one-dimensional. Metallic wires can be made as thin as 1 µm in diameter like this, but this is still far away from being one-dimensional. Lithographic processes using focused ion beams and focused electrons can produce some metal and semiconductor structures that are narrow enough to exhibit one-dimensional properties (~nanometers).

Technology today has made it possible to approach such sizes using methods of lithography as well as chemical assembly. Lithography is the top-down approach to creating confining structures as it whittles away material until only very small structures remain. Chemical assembly is the "bottom-up" approach, and it forms the structure through chemical reactions. The two approaches offer very different properties to the nanoscale structure created, both in terms of atomic ordering and control over object placement.

To achieve "one-dimensionality" does the wire puller in Figure 1.2 have to draw the wire so extensively that it is finally to become a monatomic chain? Well, the Fermi wavelength, a fundamental property of the carrier electron responsible for conductivity, becomes relevant when discussing the eigenstates of all the electrons of the structure. If electrons are confined in a box, quantum mechanics tells us that the electrons can have only discrete values of kinetic energy. The energetic spacing of the eigenvalues depends on the dimensions of the box - the smaller the box the larger the spacing (Figure 1.3):

Figure 1.3 Electrons in small and large boxes and energy spacing of the eigenstates. This is an example of...