Graphene is a two-dimensional form of crystalline carbon, either a single layer of carbon atoms forming a hexagonal lattice or several coupled layers of this honeycomb structure [1, 2]. The word graphene, when used without specifying the form, for example, bilayer graphene and multilayer graphene, usually refers to single-layer graphene. Graphene is a parent form of all graphitic structures of carbon [3, 4]: graphite, which is a three-dimensional crystal consisting of relatively weakly coupled graphene layers; carbon nanotubes, which may be represented as scrolls of graphene; and buckyballs, spherical molecules made from graphene with some hexagonal rings replaced by pentagonal rings.
The theoretical study of graphene was started in 1947 by physicist Philip Russell Wallace as a first step to understanding the electronic structure of graphite . The term graphene was introduced by chemists Hanns-Peter Boehm, Ralph Setton, and Eberhard Stumpp in 1985 as a combination of the word graphite, referring to carbon in its ordered crystalline form, and the suffix, referring to polycyclic aromatic hydrocarbons in which the carbon atoms form hexagonal, or six-sided, ring structures . In 2004 University of Manchester physicists Konstantin Novoselov and Andre Geim and colleagues isolated single-layer graphene using an extremely simple method of exfoliation from graphite . Their scotch-tape method used adhesive tape to remove the top layers from a sample of graphite and then apply the layers to a substrate material. When the tape was removed, some graphene remained on the substrate in single-layer form. In fact, derivation of graphene is not a difficult task by itself; each time someone draws with a pencil on paper, the pencil trace contains a small fraction of single-layer and multilayer graphene. The achievement of the Manchester group was not only to isolate graphene flakes but also to study their physical properties. In particular, they demonstrated that electrons in graphene have a very high mobility, which means that graphene could possibly be used in electronic applications. In 2010 Geim and Novoselov were awarded the Nobel Prize for Physics for their work . In these first experiments, the substrate for graphene was silicon naturally covered by a thin transparent layer of silicon dioxide. It turned out that single-layer graphene created an optical contrast with the silicon dioxide that was strong enough to make the graphene visible under a standard optical microscope. This visibility has two causes. First, electrons in graphene interact very strongly with photons in the visible light frequencies, absorbing about 2.3 percent of the light's intensity per atomic layer. Second, the optical contrast is strongly enhanced by interference phenomena in the silicon dioxide layer; these are the same phenomena that create rainbow colors in thin films such as soap film or oil on water.
The basic electronic structure of graphene and, as a consequence, its electric properties are very peculiar [9, 10]. By applying a gate voltage or using chemical doping by adsorbed atoms and molecules, one can create either electron or hole, a region where an electron is missing that acts as a positive electric charge, conductivity in graphene that is similar to the conductivity created in semiconductors [9, 10]. However, in most semiconductors there are certain energy levels where electrons and holes do not have allowed quantum states, and, because electrons and holes cannot occupy these levels, for certain gate voltages and types of chemical doping, the semiconductor acts as an insulator. Graphene, on the other hand, does not have an insulator state, and conductivity remains finite at any doping, including zero doping [11, 12]. Existence of this minimal conductivity for the undoped case is a striking difference between graphene and conventional semiconductors [11, 12]. Electron and hole states in graphene relevant for charge-carrier transport are similar to the states of ultra-relativistic quantum particles, that is, quantum particles moving at the speed of light, the ultimate velocity in nature, according to the theory of relativity.
The honeycomb lattice of graphene actually consists of two sublattices, designated A and B, such that each atom in sublattice A is surrounded by three atoms of sublattice B and vice versa . This simple geometrical arrangement leads to the appearance that the electrons and holes in graphene have an unusual degree of internal freedom, usually called pseudospin . In fact, making the analogy more complete, pseudospin mimics the spin, or internal angular momentum, of subatomic particles [15, 16]. Within this analogy, electrons and holes in graphene play the same role as particles and antiparticles, for example, electrons and positrons, in quantum electrodynamics [15, 16]. At the same time, however, the velocity of the electrons and holes is only about one three-hundredth the speed of light. This makes graphene a test bed for high-energy physics: some quantum relativistic effects that are hardly reachable in experiments with subatomic particles using particle accelerators have clear analogs in the physics of electrons and holes in graphene, which can be measured and studied more easily because of their lower velocity . An example is the Klein paradox, in which ultra-relativistic quantum particles, contrary to intuition, penetrate easily through very high and broad energy barriers . Therefore, graphene provides a bridge between materials science and some areas of fundamental physics, such as relativistic quantum mechanics.
There is another reason why graphene is of special interest to fundamental science: it is the first and simplest example of a two-dimensional crystal, that is, a solid material that contains just a single layer of atoms arranged in an ordered pattern . Two-dimensional systems are of huge interest not only for physics and chemistry but also for other natural sciences . In many respects, two-dimensional systems are fundamentally different from three-dimensional systems . In particular, due to very strong thermal fluctuations of atomic positions that remain correlated at large distances, long-range crystalline order cannot exist in two dimensions . Instead, only short-range order exists, and it does so only on some finite scale of characteristic length, a caveat that should be noted when graphene is called a two-dimensional crystal . For this reason, two-dimensional systems are inherently flexural, manifesting strong bending fluctuations, so that they cannot be flat and are always rippled or corrugated . Graphene, because of its relative simplicity, can be considered as a model system for studying two-dimensional physics and chemistry in general [25, 26]. Other two-dimensional crystals besides graphene can be derived by exfoliation from other multilayer crystals, for example, hexagonal boron nitride, molybdenum disulfide, or tungsten disulfide, or by chemical modification of graphene, for example, graphane, hydrogenated graphene, or fluorinated graphene.
Modern electronics, for example, integrated circuits in computer chips, are basically two-dimensional in that they use mainly the surface of semiconducting materials. Therefore, graphene and other two-dimensional materials are considered very promising for many such applications [27, 28]. Using graphene, for example, it should be possible to make transistors and other electronic devices that are much thinner than devices made of traditional materials. Many other applications have been proposed. For example, graphene, being electrically conducting, transparent, strong, and flexible, may be a prospective material for use in touch screens . Graphene also has very high thermal conductivity  and, therefore, could be used to remove heat from electronic circuits.
The field of graphene science and technology is relatively new, having emerged since Geim and Novoselov's work in 2004 [7, 8]. In the decades that followed, it remained difficult to determine which applications would prove to be the most popular . Progress depends not only on the basic science but also on the development of new ways to produce graphene on an industrial scale. Obtaining graphene by exfoliation is too expensive for mass production . Methods proposed include the formation of graphene layers by burning silicon carbide or by chemical vapor deposition of carbon on the surface of some metals such as copper or nickel [33, 34]. These methods would allow the production of samples of graphene that were macroscopically large in two dimensions, up to tens of centimeters, but still atomically thin [35, 36]. While graphene exhibits extremely high thermal conductivity, the effects of crystal length and edge states on the heat conduction properties of the two-dimensional crystal are poorly understood.
This study relates to the heat conduction properties of graphene. The effects of crystal length and edge states on the heat conduction properties of graphene were investigated by using the nonequilibrium molecular dynamics method. Computational simulations were performed using molecular dynamics to investigate the heat transport properties of graphene. The intrinsic thermal conductivity of graphene was determined to understand the characteristics of thermal transport in the nanostructured materials. The mechanism of phonon transport in graphene was discussed. The physical factors limiting heat conduction in graphene with atomically-disordered edges were provided. The objective is to gain insight into the fundamental characteristics of thermal transport in graphene. Particular emphasis is placed on the dependence of crystal length and edge states on the thermal conductivity of graphene, with an attempt to improve the heat conduction properties for carbon-based nanostructured materials.