Introduction

Investigations of two-dimensional crystals are traditionally linked to the study of surfaces. The fundamentals of surface thermodynamics were developed by Gibbs as far back as the end of the nineteenth century. Two-dimensional phases and phase transitions between them were first observed by Langmuir in the 1920s in experiments with layers of organic molecules (salts of fatty acids) on the surface of a liquid. These experiments resulted in the indirect discovery of a crystalline phase in a two-dimensional system.

Major contributions to the understanding of the two-dimensional crystals, from both the theoretical and the experimental viewpoint, date back to the 1920s and 1930s. First, Davisson and Germer succeeded in directly observing the crystalline structure of a surface in their classic experiments on the diffraction of electrons. Second, Landau and Peierls formulated a theorem on the impossibility of the existence of 2D crystals.

The rapid development of experimental research in the physics of two-dimensional systems began in the 1960s with the emergence of a multitude of methods for diagnostics of surface and other two-dimensional objects, permitting the study of these phenomena on the atomic level. The growth of interest in two-dimensional systems was, to a great extent, stimulated by the requirements of industry (microelectronics, emission electronics, the growth of crystals, catalysis, etc.).

In the late 1960s, the question of the character of ordering in two-dimensional systems with the continuous group of symmetry was reconsidered by Stanley and Kaplan, Mermin (1968), Hohenberg, Berezinsky (1971a, b), and Jancovici (1967). It was established that the logarithmic divergence of the atomic displacement fluctuations in a 2D crystal that was discovered by Landau and Peierls implies only the absence of long-range order. This divergence manifests itself in the power-law decay of the correlation function of the displacement and, as a result, in the form of the diffraction spots. In spite of this fact, the shear modulus in the 2D crystal does not equal zero.

The list of 2D crystals includes objects of radically different nature. The longest part of this list consists of the lattices formed by atoms adsorbed on a crystal surface. Two-dimensional structures on surfaces are also formed by atoms of the crystal itself, e.g., during the reconstruction of a surface. Other, less numerous classes of two-dimensional crystals include those formed by electrons trapped at the surface of liquid helium and predicted by Wigner in 1934, though for the 3D case. There are also two-dimensional lattices formed by Abrikosov vortices in superconducting films, colloidal crystals, and crystals in smectic layers and in the above mentioned Langmuir films.

The variety of these systems means that the characteristic scales of parameter values differ by at least several orders of magnitude for different classes of two-dimensional crystals. Thus, the adatom lattice periods are on the order of several angstroms, whereas the periods of colloidal crystals are on the order of several micrometers. The melting temperature of an electron lattice trapped at a helium surface is a few hundred millikelvin, whereas that of adatom lattices is several hundred kelvins.

Strong fluctuations in 2D crystals preclude many approaches conventionally used in the study of 3D crystals. The same is true of two-dimensional systems in general. Two-dimensional crystals are a special case of systems with the abelian symmetry group. Therefore, the development of the theory of 2D crystals is closely related to the evolution of two-dimensional mathematical physics. Following the solution of the Ising model by Onsager in 1944, the next major success in this area was the solution of the special case of the six-vertex model, the so-called ice model. This result has stimulated the series of works devoted to the exact solution of the general six-vertex problem, crowned by the solution of the eight-vertex model by Baxter in 1971. This model is referred to as the Baxter model. The works of Berezinsky on the model of plane rotators, also called the XY model, relate to the same period. The ideas of Berezinsky concerning the character of a phase transition in the XY model were addressed in the works by Kosterlitz and Thouless. These ideas were further developed in the works by Jose, Kadanoff, Nelson, and Kirkpatrick (1977) and Wiegman (1978) on the XY and Zn models, and also in the works by Young, Halperin, and Nelson on the melting of an isotropic two-dimensional crystal.

In the multitude of two-dimensional structures one can distinguish a small, but fundamentally important, class of free two-dimensional crystals. Among these are the freely suspended smectic films of liquid crystals. It is for these films that Moncton et al. (1979, 1982) experimentally proved in the late 1970s the existence of a two-dimensional crystal with a finite shear rigidity characterized, however, by the power-law decay of the correlation function of displacements.

For the existence of most 2D crystals an external field is needed, to ensure their two-dimensionality. Occasionally, this role is played by the matrix of a 3D crystal containing weakly interacting planes. Intercalated graphite exemplifies such a quasi-two-dimensional system.

Most 2D crystals are formed on the surface of a liquid or a crystal. Langmuir films and the Wigner crystal exemplify the systems with a liquid substrate. The correlation properties of such systems are identical to those of free two-dimensional crystals, since the surface of a liquid creates no periodic potential corrugation.

In the case of a crystal substrate the periodic potential corrugation engenders an enormous variety of structures both in adsorbed layers and on clean surfaces. The behavior of these lattices is greatly dependent on the relation of their periods to those of a substrate. For this reason, the whole set of structures found experimentally is divided into two classes: commensurate structures with periods that are multiples of the substrate periods, and incommensurate structures with at least one period incommensurate with the substrate’s period.

The theory of incommensurate structures goes back to the work by Kontorova and Frenkel in 1938. In 1949, Frank and van der Merwe demonstrated the existence of a lattice of misfit dislocations, or solitons, in a one-dimensional incommensurate chain. Incommensurate structures were discovered experimentally in the 1960s. First came helical magnetic structures, earlier predicted by Dzyaloshinsky, and then incommensurate two-dimensional crystals. Later on, incommensurate three-dimensional crystals were discovered. The physics of incommensurate crystals advanced rapidly in the 1970s and early 1980s, and this development resulted in the discovery of an entirely new type of a structure, i.e., quasicrystals, by Shechtman et al.

The spectrum of an incommensurate crystal contains an acoustical band representing the displacement of the whole crystal relative to the potential corrugation. Therefore, the two-dimensional crystal is supposed to behave as a free crystal, i.e., the correlation function should show a power-law decay. In 1983, Birgenau and colleagues have demonstrated this fact by investigating x-ray diffraction spot profiles from the incommensurate crystal formed by xenon on the basal plane of graphite. The fundamentals of the statistical mechanics of two-dimensional incommensurate crystals were developed in the works by many authors listed in Chapter 7 together with experimental results.

The statistical mechanics of commensurate structures was begun by the work of Onsager on the Ising model. At the time of its appearance, Onsager’s theory seemed somewhat rarefied. However, in the late 1960s and early 1970s its experimental realizations were discovered—first among the magnetic systems, and later among the 2D crystals. Physical realizations of many exactly solved models in two-dimensional statistical mechanics (e.g., the Potts, Baxter, and Ashkin–Teller models, the anisotropic Heisenberg model, and the hard-hexagon model) were found among commensurate two-dimensional crystals. This correspondence was established in the works by Alexander (1975) and Domany, Schick, Walker, and Griffiths (1978) and corroborated experimentally.

Berezinsky (1971a,b) discovered the important role of topological defects in the physics of 2D systems. Later on, Kosterlitz and Thouless (1973) realized that the phase transition in the XY system could be interpreted as a spontaneous creation of vortices. A similar role in the crystal’s melting is played by dislocations. Another type of topological phase transition is related to the spontaneous formation of domain walls. It is a transition from a commensurate to an incommensurate crystal. The spontaneous creation of domain walls also prevails in the phase transition in Ising’s magnetics. The topological defects in systems with a nonabelian symmetry group have a finite energy and cause a transition only at zero temperature.

We have already noted that experimental research on surfaces and theoretical research on two-dimensional mathematical physics proceeded virtually independently of each other. Therefore, despite our attempts at a unified treatment, the chapters devoted to the description of experimental methods (Chapter 2), surface structures (Chapter 3), and the relation between the structure and the properties of a surface (Chapter 12) are fairly self-contained. For consistency, everywhere in this book we...