Graphene is one of the crystalline forms of carbon related materials. In particular, graphene is similar to the crystalline structure of diamond and graphite. In fact, graphene is essentially one-atom thick layer of graphite. The term itself (“graphene”) was coined by Hanns-Peter Boehm 1 who described single-layer carbon foils (1962). The word “graphene” is a variation of the word “graphite”. Graphene is related structurally to carbon allotropes (including graphite) such as carbon nanotubes, fullerenes and even charcoal. Important feature of graphene is its one-dimensionality which sets it apart from three-dimensional forms, such as e.g. graphite.
Graphene is one of the crystalline forms of carbon related materials. In particular, graphene is similar to the crystalline structure of diamond and graphite. In fact, graphene is essentially one-atom thick layer of graphite. The term itself (“graphene”) was coined by Hanns-Peter Boehm 1 who described single-layer carbon foils (1962). The word “graphene” is a variation of the word “graphite”. Graphene is related structurally to carbon allotropes (including graphite) such as carbon nanotubes, fullerenes and even charcoal. Important feature of graphene is its one-dimensionality which sets it apart from three-dimensional forms, such as e.g. graphite.
The electronic structure of carbon gives rise to its ability to bond in many different configurations and form structures with distinctly different characteristics. This is clearly manifested in diamond and graphite, which are the two most commonly observed forms of carbon. Diamond forms when the four valence electrons in a carbon atom are sp3 hybridized (i.e., all bonds shared equally among four neighboring atoms), which results in a three-dimensional 3-diamond cubic structure. Diamond is the hardest material known to humankind due to its 3D network of carbon-carbon (C – C) bonds. It is also special by the fact that it is one of the very few materials in nature that is both electrically insulating and thermally conductive. On the other hand, graphite is the sp2 hybridized form of carbon and contains only three bonds per carbon atom. The fourth valence electron is in a delocalized state, and is consequently free to float or drift among the atoms, since it is not bound to one particular atom in the structure. This creates a planar hexagonal structure (called graphene) and gives rise to the layered structure of graphite that is composed of stacked two-dimensional (2D) graphene sheets. Graphite contains strong covalent bonds between the carbon atoms within individual graphene sheets, which gives rise to its outstanding in-plane mechanical properties. However, the van der Waal’s forces between adjacent graphene sheets in the layered structure are relatively weak and, therefore, graphite is much softer than diamond. Similar to diamond, graphite (in-plane) is a good conductor of heat, however, the free electrons present in graphite also endow it with high in-plane electrical conductivity, unlike diamond. The crystal structure of diamond and graphite is depicted schematically in Fig. 3.1.
Figure 3.1 Schematic of the atomic structure for a) graphite, showing the sp2 hybridized hexagonal lattice, and b) sp3 hybridized diamond, which consists of the 3-dimensional diamond cubic lattice.
As a result of its unique two-dimensional crystal structure and ultra-strong sp2 carbon bonding system, graphene offers a promising blend of electrical, thermal, optical and mechanical properties that paves the way to a variety of possible applications. The elastic modulus of an individual graphene sheet was predicted 1 to be ~ 1 TPa (or 1,000 GPa). The data has been confirmed 1 by atomic force microscopy (AFM) – based indentation experiments performed on suspended graphene. The exceptionally high modulus of graphene, along with its low density (~ 1 – 2 g/cm3), means that the specific modulus (i.e., modulus normalized by density) of graphene far exceeds that of all other structural materials, including aluminum, titanium, or steel. In addition to its very large elastic modulus, graphene also displays a fracture strength of ~ 125 GPa1, which is higher than most structural materials demonstrate. Graphene has a very interesting electronic band structure. It may be called a semimetal with zero electronic band gap; the local density of states at the Fermi level is also zero and conduction is only possible by the thermal excitation of electrons 1 . However, an energy gap may be engineered in graphene’s band structure in different ways. The possible methods are based on the breaking of graphene’s lattice symmetry, such as defect generation 1 , water adsorption 1 , applied bias 1 , and interaction with gases (e.g., nitrogen dioxide or ammonia). Among other outstanding properties of graphene that have been reported are exceptionally high values of its in-plane thermal conductivity (~ 5,000 Wm−1K−1), charge carrier mobility (200,000 cm2V−1s−1), and specific surface area (2,630 m2g−1) (i.e. the total surface area per unit of mass), plus remarkable features such as the quantum Hall effect, spin resolved quantum interference, ballistic electron transport, and bipolar super-current and others. It should, however, be noted that the exceptional thermal and electronic properties of graphene (that are listed in literature) are usually true only for the free-standing graphene.
In practice, the achieved 30-inch, one-atom-thick graphene sample certainly would be so unstable that it would have to be placed on some surface. This is the real question as to whether graphene is a crystal. If we imagine the honeycomb sheet as unsupported, we realize it is very susceptible to being bent out of its flat planar condition. The chemical bonds (“π-bonds” between two 2pz electrons) tend to return it to a flat planar condition, but the resulting force is too weak against flexing motion. It is like a sheet of paper, in being flexible but, unlike a sheet of paper, it retains a weak restoring force toward a perfectly planar condition. In the crystal form, inherent in two dimensions (2D) (embedded in three-dimensional space), we have long-wavelength flexural phonons that allow large root-mean-square (rms) fluctuational displacements much larger than a lattice constant. The sheet size determines how unstable it is. The statement, whether the graphene sheet is crystalline can be debated. By formal definition, long-range order does not take place, but in practice the local distortions may be so small, so that it is still acceptable to consider the sample a crystal, even if it is slightly distorted. For graphene in practice, the out-of-plane deflections pose the main problem: as to whether the system is crystalline. In addition, there are more subtle points, mostly of academic interest. One is that 2D array, even if arbitrarily kept absolutely planar, cannot have long-range (infinite) order except at T = 0. In this case, the planarity would have to be imposed without transverse pinning. The closest system of this type may be electron crystals on the surface of liquid helium. An infinitely large 2D array would exhibit, at any finite temperature, large absolute in-plane motions (but without tangibly affecting local inter-atom distances). The above motion may cause smearing of diffraction patterns of electron or x-rays on a large sample.
Since the phonon wavelengths (now in 2D) involved are large, local regions may move without disrupting of the local order. In this case, the melting point connected with local order is not influenced by changes in cohesive energy of the system. While there had been earlier suggestions that the single planes of graphite might be extracted for individual study (contrary to a theoretical literature that suggested that crystals in two dimensions should not be stable), Novoselov et al. (2004, 2005) 2 were the first to demonstrate that such samples were viable, and indeed represented a new clan of 2D materials with useful properties and potential applications. (Hints toward isolating single layers had earlier been given by Boehm et al., Van Bommel et al. (1975), Forbeaux et al. (1998) among others) 3 . On small size scales, (approximately from 10 nm to 10 µm), the graphene array of carbon atoms is “crystalline” and has sufficient local order to provide electronic behavior as predicted by calculations based on an infinite 2D array. Micrometer-size samples of graphene show some of the best electron mobility values ever measured. In microscopy, on scales 10 nm to 1 µm, it sometimes appears that the atoms are not entirely planar, and undulate slightly out of plane. It has been suggested that such “waves” are intrinsic due to the usual response of the thin membrane to inevitable deformation from its mounting, or as a result of adsorbed molecules, since in graphene every carbon atom is exposed. Monolayer graphene is strong (the space per layer in graphite is 0.34 nm that is widely quoted as the nominal thickness of the graphene layer). An equivalent elastic thickness of graphene, closer to the actual thickness is about 0.1 nm and t ~ 0.34 nm. All but the shortest samples are extremely “soft” and may be bent with a small transverse force. This can be understood from the definition of classical “spring constant K” (the spring constant K is a macroscopic dimension-related engineering quantity in SI units of N/m). It is related to the “bending rigidity” or “rigidity” k = Yt3, a microscopic property usually measured in eV which is about 1 eV in graphene. The Young’s modulus Y, an engineering quantity, is defined as pressure. It is about 1012 N/m2 = 1 TPa for graphene. The rigidity k has units of energy, as force times distance. We can see that the rigidity k of graphene, by virtue of the minimal atomic value of thickness t, is the lowest of any possible material. With extension of a chemical bond, the spring constant K relates to the bond energy E as K = d2E/dx2. For deflection x of a cantilever of width ω, thickness t and length L on (with Young’s modulus Y) under a transverse force F: F = -Kx. Since K ~ Yωt3/L3, with t near a single atom size, we can see that graphene, (in spite of a large value of Young’s modulus, Y ~ 1 TPa), is the softest material against transverse deflection. Graphene length L, width ω and thickness t, quantitatively bend and vibrate as predicted by classical mechanics formulas. For example, the spring constant K defined for deflection and applied force at the center of a rectangle clamped on two sides depends strongly on the linear dimensions as K = 32Yωt3/L3. A square of graphene, of size L = ω = 10 nm, from the above expression, yields K = 12.6 N/m, while a square of size 10 µm gives K = 12.6*10−6 N/m. If the sample is short, approaching atomic dimensions, the spring constant is large and the object appears to be rigid. For example, the spring constant of a graphene square (ten benzene molecules along a side) as it is bent is ~ 156 N/m, using the above expression. At the same time, the spring constant of a carbon monoxide (CO) molecule (in extension), deduced from its measured vibration at 64.3 THz, is known to be 1860 N/m. The next quantity in the graphene characteristics is Yt, a 2D rigidity that has a value of about 330 N/m. If the graphene sample is longer than a few micrometers, with the spring constant K of a square sample reduced to 1/L2, the sample is exceedingly soft.
Accordingly, graphene, on a micrometer-size scale, adapts to any surface under the influence of attractive van der Waals forces. In an electron micrograph, graphene on a substrate appears adherent, more like a flat piece of cloth or “membrane” than a piece of a cardboard, and quite unlike a 10-inch diameter silicon wafer 4 . Graphite and diamond are grown in the depths of the earth at high temperature, but graphene 2D “crystals” cannot, at present, be grown from a melt, similar to silicon. Graphite crystals can only be obtained by extraction from an existing crystal of graphite, or by being grown epitaxially on a suitable surface such as SiC or catalytically on Cu or Ni from a carbon-bearing gas such as methane. Notably, graphene is an excellent electronic conductor, much like a semimetal, but with conical rather than parabolic electron energy bands near the Fermi energy and with a characteristic linear dependence of energy on crystal momentum, k = p/h; i.e. E = “pc” = c*hk. These electrons move like photons, at speed c*106 m/s and with vanishing effective mass. An explanation results from the band theory for a particular crystal lattice. This aspect also presents a new paradigm in the field of condensed matter physics. Not only is graphene structure comes the closest to a two-dimensional (2D) self-supporting material, but it also has charge carriers moving in a different way, as if their mass were zero. The physics of the phenomenon also implies that “back-scattering” is “forbidden” and gives much larger carrier mobility.
In the real world of atoms, no material can be mathematically two dimensional: the probability distribution P(x, y, z) must extend in the z-direction by at least one Bohr radius 1 . The well-known examples of 2D subsystems of particles are electrons on the surface of liquid helium and the “2 – DEG” two-dimensional electron gases engineered into prospective semiconductor devices. The latter useful electron systems are supported by quantum well heterostructures. The remarkable difference, in graphene, is that there is no external supporting system: the layer of carbon atoms is the mechanical support, as well as the medium exhibiting light-like propagation of electrons. The above situation was not entirely clear before the discoveries of Geim and Novoselov: indeed the existence of free-standing graphene layers with new and superior electronic properties was a surprise, worthy of a Nobel Prize in Physics. Other one-layer materials include BN(BN)n(C2)m, with n, m, integers; MoS2, TaS2, NbSe2 and the superconductor Bi2Sr2CaCu2Ox, although the last is seven atoms thick (Novoselov et al., 2005) 5 . So Geim and Novoselov, in fact, confirmed the practical reality of a new class of 2D locally crystalline materials.
The binding energy of a crystal, an extended periodic array of atoms, for temperatures below melting temperatures, TM, is a subject of solid state physics 6 . The methods of this discipline do not always predict binding of an infinite 2D crystal. Indeed, thin layers of many substances are found to break up into small pieces (or “islands”) as their thickness is reduced, especially if the attraction of an atom to substrate exceeds the attraction atom-to-atom. This island breakup definitely does not occur with graphene: on the contrary, graphene is found to be among the strongest known materials with force applied. Recently, tenth-millimeter scale sheets of one-atom-thick graphene have been studied as elastic beams and sheets. At a lattice constant of 0.246 nm, a 20 μm graphene sheet (80,000 unit cells) looks flat, if suspended across a trench, but may bend in response to van der Waals forces from the mounting. In some cases, 10 nm-scale “waves” or “ripples” of ~ 1 nm amplitude have been influenced judging by transmission electron microscope measurements, with a likely origin in a combination of molecular surface adsorbates and mounting strain. Notwithstanding some unclear physical mechanism experiments confirm the fact; these “crystals” are large enough to be useful under many circumstances. Molecules have vibrations: in an extended crystal these are called phonons. The vibrational motions of molecules are 3D in nature and any real 2D crystal should have a useful notation (for a “real 2D crystal” is “2D-3”) meaning that motion into the third dimension is available. A “pure” 2D system is one, (like electrons on the surface of liquid helium), where no motion into the third dimension is allowed. The z-motion, is represented by a single quantum state. We take the electron system inside graphene to be “pure 2D” as confined by the graphene lattice, even though that lattice may slightly undulate or flex into the third dimension. The lattice has vibrational motion in the z-direction (called flexural). In an extended real 2D sample the flexural motion extends to low frequency and large amplitude, at any finite temperature T. Even when restricted completely to planar motion, the methods of solid state physics have predicted that thermal vibrations at any finite temperature lead to excessive transverse motion and destroy long-range (as distinct from short-range) order 4 .
Transmission electron microscopy (TEM) reveals a honeycomb lattice. The bond length of the crystal is approximately 0.142 nm. The graphene layers form a stack, the interplanar distance is about 0.335 nm. The lattice has characteristic “rippling” of the graphene sheet, with amplitude of about one nanometer. The ripples may be attributed to the instability of two-dimensional crystals or due to contamination 7 , 8 . Ripples of graphene on thesubstrates may also be explained by conformation of graphene to the substrate which, in this case, is rather an extrinsic than an intrinsic effect 9 .
Like some single-wall carbon nanostructures, graphene exhibits (002) plane preferable stacking. On the other hand, the unlayered graphene tends to form (h k o) rings that were reported to be in presolar graphite onions 10 . Naturally – occurring unlayered solidified graphene exits in the universe. In particular, unlayered graphene was found in the center of carbon spheres that were extracted from the Australian meteorite. Man-made counterparts of the natural graphene include bulk assemblies of single graphene sheet carbon molecules. Since carbon has a higher melting temperature than that of any other solid substance, it is a valuable quality for graphene mechanical applications. That is where the term “presolar graphite onions” comes from. The hexagonal shape of pure carbon atoms easily accepts carbon atom if any vacancies exist. Such an effect is achievable during bombardment with pure carbon atoms, or in case of being exposed to, e.g., hydrocarbons 11 .
As it was already mentioned, intrinsic graphene does not belong to either metals or semiconductors. It may be classified as a “zero band-gap” semiconductor (or semimetal as it was suggested above). Its electronic structure has a number of features that drew attention of the early researchers of the material. The early studies (as early as 1947) 12 revealed zero effective mass for electrons and holes, in particular, that the energy momentum relation (dispersion relation) is linear in the two-dimensional hexagonal Brillouin zone. The electrons and holes near the six points of the Brillouin zone behave like relativistic particles (Dirac equations for spin ½ particles) 13 . The Dirac points are located in the six corners of the Brillouin zone, where the electrons and holes are called Dirac fermions. The electron linear dispersion is given as:
where, the Fermi velocity (~ ); k is the wave vector which is measured from the Dirac points. In the Dirac points, the energy is zero 14 . For the electron and hole mobility is very high, reaching 15,000 (ref. 13). Between the temperatures of 10 K and 100 K the mobility is almost constant which means that the scattering is caused mainly by defects. The room temperature mobility has the limit of 200,000 for intrinsic graphene. The resistivity is at a carrier density of . It is still bigger than the resistivity of silver (which has the lowest resistivity of a solid at room temperature) 15 . In practice, the graphene resistivity is lower, e.g. up to 40,000 on substrates 16 .
The electrons occupy theatomic orbitals (Fig. 3.2). As a tetravalent element, only its four exterior electrons form covalent chemical bonds. The ground state in particular is given in a) forming bonds with other atoms. Carbon gives out one of the 2s electrons to the vacant 2pz orbit. As a result, a hybrid orbital forms. For example, diamond 2s-energy level and the three 2p level form four -orbitals that have one electron in any one of them (Fig. 3.2 b). The four -orbitals form imaginary tetrahedron. The -orbitals of different atoms overlap thus creating the 3D diamond structure. The strong binding energy of the C-C bonds gives the diamond structure its hardness. In hybridization, only two of the three 2p-orbitals participate (Fig. 3.2 c). The remaining 2p-orbitals are situated in the X-Y plane at angles. The -orbitals are perpendicular to the above 2p-orbitals. Thus, the formed covalent in-plane bonds define the hexagonal structure of the graphite. The in-plane -bonds within the graphene layers are even stronger than the C-C bonds in -hybridized diamond. At the same time, the interplane -bonds (formed by the remaining 2p-orbitals) posses a substantially lower binding energy. It causes an easy shearing of graphite along the layer plane. The measurements performed by Haering R.R. still in 1958 give the following parameters: a single graphene layer has a lattice constant ; where The distance between two adjacent layers is 3.35 in AB-stacked graphite.
Figure 3.2 Atomic orbital diagram of a carbon atom
The presence of H (hydrogen), O (oxygen), or other C (carbon) atoms is advantageous to excite one electron from the 2s to the third 2p orbital in order to form covalent bonds with the other atoms. In such a scenario, the gain in energy exceeds 4 eV which is necessary for the electron excitation. Thus, in the excited state, we have four equivalent quantum-mechanical states, I2s>, I2px>, I2py>, and I2pz>. It is important to note that a quantum-mechanical superposition of the state I2s> with n I2pj> states is called spn hybridization that is substantial in covalent carbon bonds.
The considered here sp1 hybridization (often called just “sp hybridization” for simplicity) mixes I2s> state with one of the 2p orbitals. We’ll consider the I2px> state as an example. A combination of the symmetric and anti-symmetric combinations produces:
In this case, the other states, I2py> and I2pz>
The electronic density of the hybridized orbitals form drop-like configuration as shown in Fig. 3.3. The drops are elongated in the +x (−x) direction for the Isp+> (Isp−>) states. The described above process of hybridization plays a role, for example, in the formation of the acetylene molecule(as illustrated in Fig. 3.3 (b)). The overlapping sp1 orbitals of the two carbon atoms form a strong covalent σ-bond. The 2p orbitals then participate in the formation of the two additional π-bonds that are weaker than the σ-bond.
Figure 3.3 a) sp1 hybridization. On the r.h.s. the hybridized orbitals formed from electronic densities of the I2s> and I2px>. b) Acetylene molecules ( H − C ≡ C − H ).
In sp1 hybridization, the s-orbital and one of the p – orbitals from carbon’s second energy level are combined together to form two hybrid orbitals. These hybrid orbitals form a straight line. The angle between the orbitals is 1800. The orbitals are opposite from one another with respect to the center of the carbon atom. Since this type of sp hybridization uses only one of the p – orbitals, these are still two p – orbitals left which the carbon can use. The p – orbitals have 900 – angle between them and with respect to the line formed by the hybrid orbitals. The described type of hybridization takes place when a carbon atom is bonded to two other atoms.
If we have a superposition of the 2s and two 2p orbitals (which may be chosen as the I2px> and I2py> states), we can obtain the planar sp2 hybridization. Their orbitals are in the x-y plane and have mutual 120° angles (Fig. 3.4a).
Figure 3.4 Hybridization of carbon
The third (unhybridized) 2pz orbital is perpendicular to the plane. One characteristic example for the above hybridization is the benzene molecule which is a hexagon with carbon atoms at the corners connected by σ bonds (Fig. 3.4 b). Each carbon atom has a covalent bond with one of the hydrogen atoms situated in a star-like arrangement. In addition to the above 6 σ-bonds, the remaining 2pz orbitals form 3 π-bonds. Double bonds alternate with single σ bonds in the hexagon. Since the double bond is stronger than the single one, the hexagon is not uniform. Also, the lengths of the bonds are different: a double bond has a carbon-carbon distance of 0.135 nm and 0.147 nm for a single σ-bond. Thus, a graphene sheet may be simply considered as tilted benzene hexagons with the hydrogen atoms replaced by carbon atoms to form a carbon hexagon (Fig. 3.4 c) – graphene honeycomb lattice. The honeycomb lattice due to the sp2 hybridization is not a Bravais lattice (the neighboring sites are not equivalent).
Please note that the distance a = 0.142 nm is an average of the above-mentioned distances of a = 0.135 nm and 0.147 nm. Fig. 3.5 depicts the triangular Bravais lattice, taking it as a basis, we can visualize the honeycomb lattice with a two-atom basis. The average distance between carbon atoms is 0.142 nm which is the average of the single (C-C) and double (C=C) covalent σ bonds. The following vectors connect a site as the A sublattice with the nearest neighbors on the sublattice B:
and the basis vectors of the triangular Bravais lattice:
The reciprocal lattice in Fig. 3.6 is defined with respect to the triangular Bravais lattice. The vectors of the lattice are:
Figure 3.5 Graphene’s honeycomb lattice. The vectors a → 1 and a → 2 are basis vectors of the triangular Bravais lattice.
Figure 3.6 The reciprocal lattice for the graphene’s honeycomb lattice.
All sites of the reciprocal lattice represent equivalent wave vectors. If a quantum-mechanical electron wave packet or vibrational lattice excitation propagate in the lattice, wave vector difference equals a reciprocal lattice vector’s phase with a 2π factor:
The first Brillouin zone (the shaded region and thick lines of the hexagon) represents a set of points in the reciprocal space which may not be connected to one another by a reciprocal lattice vector. In the propagation of any type of wave motion through a crystal lattice, the frequency is a periodic function of wave vector k. This function may be complicated by being multi-valued; that is, it may have more than one branch. Discontinuities may also occur. In order to simplify the treatment of wave motion in a crystal, a zone in k – space is defined which forms the fundamental periodic region, such that the frequency or energy for a k outside this region may be determined from one of those in it. This region is known as the Brillouin zone (as it was mentioned above). It is also called the first Brillouin zone. It is usually possible to restrict attention to k values inside the zone. Discontinuities occur only on the zone’s boundaries. The Γ point is the location for long wavelength excitations. Further, there are six corners of the first Brillouin zone which consists of non-equivalent points K and K` represented by the vectors:
The four other corners may be connected by reciprocal lattice vectors. These points at these corners play a role in defining the electronic properties of graphene. The points K and K` are centers of low-energy excitations. The form of the first Brillouin zone is an intrinsic property of the Bravais lattice regardless of the presence of other atoms in the unit cell. The rest of the points in the reciprocal lattice in Fig. 3.6 are shown for completeness.
The above discussion has been concerned with a single-layer graphene. If there are several layers of carbon atoms, the properties will be different from single-layer electronic properties. Only single-layer graphene (SLG) and bi-layer graphene (BLG) correspond to semiconductors with a zero gap with only one type of electrons and holes. If the number of layers is fewer than 10, the conduction and valence bands overlap which results in charge carriers 16 . If the number of graphene layers exceeds 10, we deal with a thin graphite film.
Graphene oxide (GO) and reduced graphene oxide (rGO) are alternative versions to graphene. In the case of GO, diffraction patterns reveal a nonlinear behavior. The nonlinearity may be attributed to so-called “short-wavelength” ripples or distortions comparable to inter-atomic distance which is ~ 10 % of the carbon-carbon distance 18 . The distortions are connected with the strain in the lattice. This may indicate fundamental structural and topographical differences between different layers. Using atomic force microscopy (AFM), direct visualization of graphene rippling is achievable. In particular, GO is substantially rougher when it is free-standing in comparison with the cases when GO is deposited on (an atomic scale) smooth surfaces. The observed distortions were on the order of 10 nm (approximately the size of the AFM tip).
As it was mentioned earlier, the properties of graphene depend substantially on the number of atomic layers of the material. Usually, the following classification is adopted: a single-layer, a bi-layer, and a few layer graphene (the last one implies the number of layers should not exceed 10). The second aspect that has substantial influence on the electronic band structure of graphene is how the graphene layers are stacked 6 .
In AB-Bernal stacking the third layer is above the first layer. In Rhombohedral ABC stacking (Fig. 3.7) the third layer is shifted with respect to the first and second layers. The fourth layer is located, in this case, above the first layer. For AB-Bernal stacking (Fig. 3.7), alternate layers have the same projections on the X-Y plane. The distance between the layers is 3.35. In ABC-stacked Rhombohedral graphite, the third layer is shifted with respect to the first and second layers. The fourth layer and the first layer have the same projection on the base layer. The separation between the layers becomes 3.37 now 6 . The adjacent graphene layers are parallel. They may contain rotational stacking faults (since they can rotate relative to each other without a preferred orientation). If multilayer graphene is grown on the carbon-terminated face of SiC, it obtains a high density of rotational disorder 6 . Yet, these multilayered structures are of interest because introduction of rotational stacking faults into the ABC may cause a separation of adjacent layers.
Figure 3.7 AB-Bernal stacking and ABC Rhombohedral stacking
Near the Dirac points, graphene has a minimum conductivity equal approximately to. Taking into account the zero carrier density at the above points, the origin of the conductivity is not clear at the moment. The mentioned before rippling and the presence of defects may explain the local density of carriers. Many measurements confirm the factor of or even greater depending on the impurity concentration 19 .
The chemical dopants also have influence on the carrier mobility in graphene. However, the influence is less than could be expected: even for chemical dopant concentration ofthere is no observable change in the carrier mobility. The noticeable results have been reported by Chen, et al 19 . They found that potassium ions could reduce the mobility 20 times as long as the potassium was present in the graphene. The band gap of graphene can be changed within the limits of ~ 0 to 0.25 eV using a dual gate bi-layer graphene FET (field effect transistor) at room temperature. The optical radiation frequency to produce the above effect is about 5 μm (Zhang Y. et al).
The double-layer graphene (double-GL) structures have been studied and fabricated recently 20 . The structure consists of two GLs separated by a thin layer. Such a structure deserves a considerable attention due to its potential applications for optical transparent transistor circuits, THz lasers, THz detectors, frequency multipliers, and THz photomixers. In particular, the double GL structures exhibit tunneling or thermionic conductance, the effects that be utilized in resonant detectors and photomixers 21 . The experiments prove that the inter-GL resonant tunneling in the double-GL structures results in inter-GL negative differentiated conductivity - valuable property for the creation of a new generation of transistors with multi-valued current-voltage characteristics.
One-atom-thick layer is almost invisible to the naked eye since onlyof white light is actually absorbed. Where = fine structure constant (fine structure constant is the coupling constant characterizing the strength of the electromagnetic interaction). However, even such a low percentage is high enough in comparison to other materials this thin. This opacity comes from the unusually low-energy structure that a graphene monolayer possesses. The electron and hole conical bands connect with each other at the Dirac points. The electronic structure is different from more usual quadratic massive bands 22 . The measurement of the fine structure constant nevertheless remains rather difficult, often with insufficient precision. The optical conductance can be calculated using the traditional Fresnel equations for the thin-film limit. The basis for the calculation is provided by the Slonczewski-Weiss-McClure (SWMcC) graphene band model.
It has been reported that the optical response of graphene (graphene nanoribbons, in particular) may extend into THz range 23 by an applied magnetic field as well as graphene may change its properties for both linear and ultrafast regimes 24 . A practical application example may be a graphene-based Bragg grating (which is a one-dimensional photonic crystal) that has been announced as capable of excitation of surface electromagnetic waves in a periodic structure (such as a grating). As a light source a 633 nm He-Ne laser may be used 25 . In addition, not only the magnetic field but an electronic current can be used to influence the optical response. Another effect is saturable absorption in the THz and microwave ranges. It occurs when the input optical intensity exceeds the threshold value. The saturable absorption is a nonlinear effect. Graphene may be saturated under strong excitation due to its zero-band gap. This saturation effect relates to mode-locking of fiber lasers. The full-band locking is achieved by a graphene-based saturable absorber. Under more intensive laser radiation, there may be a nonlinear phase shift (optical nonlinear Kerr effect). Under open and closed aperture z-scan measurement, graphene may be able to produce a big nonlinear Kerr coefficient of, which is almost times larger than that for bulk dielectrics (in nonlinear optics, a z-scan measurement is used to measure the nonlinear index , Kerr nonlinearity and the nonlinear absorption coefficient using “closed” and “open”). Such an effect opens new possibilities for graphene-based nonlinear Kerr photonics, e.g. solition in graphene 26 . All of the above-mentioned phenomena may contribute to different devices and their improvements: fiber laser, mode-locking (full-band mode locking is achieved by graphene-based saturable absorber), microwave saturable absorber, polarizers, modulators in the microwave range, and broad-band wireless access networks to name a few27.
Graphene’s thermal properties are different from those of other carbon materials, in particular, nanotubes or diamond. The room temperature thermal conductivity is within the limits of. These measurements were performed by non-contact optics. The isotropic composition has an influence on the graphene thermal non-contact optics development. The isotropically pure 12G graphene has a higher conductivity than the natural graphene occurring 99:1 ratio, or 50:50 isotropic ratio. The thermal conductance of graphene is isotropic, provided it is ballistic. In general, the conductance is not ballistic, i.e. there is scattering by impurities, defects, or by the atoms or molecules composing the medium. In our case, the ballistic conductance (conduction or transport) may exist due to the medium of negligible electrical resistivity and scattering. Thermal conduction can be phonon or electron dominated. The nature of the thermal conduction depends on whether the applied voltage causes an increase in electronic contribution (caused by a Fermi energy shift greater than ) over the phonon contribution at low temperatures.
In graphite, the vertical (the c-axis, out of plane) thermal conductivity is more than a hundred times smaller than basal (i.e. parallel to the lateral or horizontal axis) plane thermal (of over 1,000). Also, the ballistic thermal conductance of graphene is lower than that of other carbon structures 28 . As the temperature decreases, the resistivity’s magnitude diverges from a power-law (i.e. a functional relationship between two quantities where one quantity varies as a power of another) behavior and becomes as high as several megoohms per square at about 20 mK as opposed to the commonly observed saturation of the conductivity. With an applied perpendicular field, the graphene layer remains insulating with direct transitions to the broken-valley-symmetry, quantum Hall state, which means that the insulating behavior at zero magnetic field is a result of the broken-valley-symmetry. At zero magnetic field, the disorder landscape dominates 29 . Recently, the disorder influence is reported to be reduced by keeping the graphene sheet on automatically supplied flat boron nitride (BN) 30 . Anomalous patterns, in this case, were seen as to be caused by a close alignment of the graphene and boron lattices.
The nature of the conductivity at the charge-neutrality point,has been discussed in connection with graphene-based devices fabrication. As mentioned earlier, theory of ballistic transport in graphene yields a value of Experimentally, however, the value of is between 2 and 18 . depends on the carrier density produced by static charges close to the graphene surface. However, the conductivity saturation in (particularly in suspended graphene) still occurs at low temperature and remained higher than . As it was mentioned earlier, graphene exhibits the qualities of a two-dimensional material, exhibiting a high crystal quality (electrons can move distances of less than a micron without scattering), on the other hand, two dimensional crystals cannot exist in the free state. This may be avoided if graphene sheets are supported by a bulk substrate or embedded in a three-dimensional matrix. Individual graphene sheets may be freely suspended on a microfabricated scaffold in the air or vacuum. This fact, brings us to the concept of “suspended graphene”. Further work was done by screening of potential fluctuations by placing an additional sheet of doped graphene nearby the tested sheet. In this case 6 , instead of saturating at the values close to , decreased with a power law dependence temperature dependence , where α = 2 for the most insulating graphene sheets (samples) for the temperatures as low as 4 K. In addition, a strong magneto-resistance was observed at the temperatures higher than 10 K which was attributed to weak localization. The implication was that if the sample was absolutely clean, then the graphene may be an Anderson insulator (it implies the possibility of electron localization inside a semiconductor, provided that the degree of randomness of the impurities or defects is sufficiently large). Also, it was concluded that this temperature dependence is explained by increasing order in the sample crystal and, as a result, the sample becomes more insulating. Hence, is the conduction temperature dependence in the presence of electron and hole puddles 31 . Obviously, complete understanding of graphene conductivity is still lacking at the present moment. The temperature dependence of the graphene resistance can be used to measure the entropy level in graphene samples. Less contaminated samples show a higher resistivity which increases with the temperature drop. The saturation takes place when is smaller than the Fermi energy’s fluctuation 32 .
Graphene has three acoustic phonon modes: two in-plane modes (LA, TA) (that have a linear dispersion relation) and the out-of-plane mode (ZA) (that has a quadratic dispersion relation). At low temperatures, the thermal conductivity T2 contribution of the linear modes is dominated by out-of-plane modes, T1.5. In addition, at low temperatures, the Gruneisen parameters dominate the thermal conductivity: the thermal coefficient is negative under such conditions. Some of the Gruneisen parameters may be negative for the graphene (most optical modes with positive Gruneisen parameters are still not excited at low temperatures). Please note that Gruneisen parameter,, (named after Eduard Gruneisen) determines how the changing volume of a crystal lattice influences its vibrational properties and, consequently, how the changing temperature influences the size or dynamics of the lattice. Thus, the thermal expansion coefficient is negative at low temperatures 25 . The negative Gruneisen parameters correspond to the lowest transversal acoustic (ZA) modes. In-plane frequencies in x-y (in-plane) direction are greater than in z-direction since the atoms are more restricted to move in z-direction. This phenomenon is similar to the string stretching and vibrating where the latter takes place mostly along the string. This “membrane effect” was predicted in 1952 by I.M. Lifshitz 28 .
At the moment, the graphene is one of the strongest materials from the mechanical point of view. If it were possible to experimentally compare the breaking strength of steel and graphene films of the same thickness, the graphene would exhibit over a 100 times greater breaking strength than steel. For an industrial purpose in particular, or for many other practical applications, the separation a few-atom layer sheets from graphite is difficult. Technological innovations are, thus, necessary before graphene becomes an economically usable material for series- production devices. Graphene weighs only(which is approximately 0.001 % of the weight of 1 m2 of paper). Since graphene sheet (or graphene paper as it is sometimes called) can undergo different reshaping from its original state it gives the promise of future manufacturing out of graphene. The easiness of shaping and firmness of thin graphene layers may facilitate making lighter vehicles, airplanes, etc. that use less fuel and generate less pollution. As carbon-based fibers are taking place of the aircraft materials so could graphene in time 33 .
According to the Mermin-Wagner theorem, the amplitude of long-wave fluctuations in regions of graphene grows logarithmically with the scale of a 2-D structure and is unbounded in structure of infinite size. The implication is that long-range fluctuations can be created with little energy since these fluctuations increase the entropy of the structure. However, the local deformations and elastic strain are negligibly affected for relative displacement on a large scale. Thermal fluctuations can cause ripples in graphene. Also, it is believed that a sufficiently large (2-D) sheet of graphene will deform to become a fluctuating 3-D structure. The above properties cast doubt on the notion that graphene is a genuine 2-D structure 29 .
The transverse conductivity in graphene depends on a magnetic field which is perpendicular to the current. The quantum Hall effect can usually be observed in Si or GaAs solids almost absolutely devoid of contamination or foreign additives and at very low temperatures (around 3 K). The quantization of the Hall conductivityimplies a multiplication of the basic quantity , where e is the charge of the electron and h is Plank’s constant. Graphene exhibits an anomalous quantum Hall effect which can be measured 34 . The conductivity is , where N is the “Landau level” index (see Fig. 3.8). The double valley and double spin degeneracies give the factor of 4.
Figure 3.8 Quantum effect in graphene
Thus, the usual sequence of steps is shifted by ½ and with an additional factor of 4. The described anomalous behavior is a result of Dirac electrons in graphene with no mass. In general, Dirac electrons relate to the “Dirac sea” theoretical model of the vacuum as an infinite sea with negative energy. The theory served to explain the anomalous negative-energy quantum states predicted by Dirac equation for relativistic electrons. When exposed to a magnetic field the half-filled Landau level (in quantum mechanics, the quantization of the cyclotron orbits of charged particles in magnetic fields results in the charged particles being able to occupy only orbits with discrete energy levels known as “Landau levels”) corresponds to an energy level exactly at the Dirac points that are defined by the Atiyah-Singer theorem (in differential geometry the Atiyah-Singer theorem states that for an elliptic differential operator on a compact manifold (which is a type of topological space without a boundary)), the analytical index is equal to the topological index (which comes from the topological data). As a result, we have + ½ in the Hall conductivity. The quantum Hall effect is also valid for bilayer graphene, however, with only one anomality, i.e.. In this case, the first plateau at N=0 is absent, meaning that bilayer graphene exhibits metallic qualities at the neutrality point 5 .
The longitudinal resistance of graphene is maximum (minimum for normal metals) for integral Landau filling factors in measurements of the Shubnikov-DeHaas oscillations (unlike in bulk materials, in a 2-dimensional system the electrons can only move in one plane and may not travel perpendicular to this plane). Such measurements show a phase shift (equal to π) known as Berry’s phase. In classical and quantum mechanics the Berry’s phase is a phase acquired over the course of a cycle, when the system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The Berry’s phase occurs when the amplitude and phase are changed simultaneously but very slowly (adiabatically) and eventually are brought back to the initial configuration. Close to the Diracs points, the Berry’s phase is explained by the zero carrier mass. However, the carriers still have a non-zero cyclotron mass 35 .
With sufficiently strong magnetic fields (more than 10 Teslas), additional plateaus of the Hall conductivity appear at. The Hall effect was observed also at and partially at 1/3 33 . The quantum Hall effect is responsible for eliminating the degeneracy of the Landau levels with . The degeneracy is fourfold (two valley and two spin degrees of freedom).
Graphene’s physical properties make it a promising material to achieve high laser efficiency at room temperature. In particular, plasmonic dynamics exploit the optical properties of graphene to design a new type of a solid state laser in the THz range 36 . THz solid-state lasers are commonly used at the present for THz imaging and spectroscopy. Their efficiency, however, is low and they do not work at room temperature. Notwithstanding the designers’ efforts in the past decade or so to produce a compact, tunable and coherent THz source, there are still no commercially available sources that cover the whole THz range. In a photonic device, such as a QCL (semiconductor quantum cascade laser), a decrease in operating frequency takes place by scaling down the energy of transition from the interband level (where electrons and holes transfer between two adjacent energy state bands in a semiconductor) to the inter-sub-band. The QCLs, however, produce substantial thermal noise at room temperature making them unusable as efficient THz sources. In the conventional field-effect transistor (FET), in order to increase the operating frequency, the distance between the source and the drain sides of the electron channel is scaled down but this reduces the output power of the devices.
In order to overcome the described difficulties, a graphene structure with unique transport of carrier, and unique optical properties has been proposed 1 . The conduction and valence bands are symmetrical around the edges of the Brillouin zone. The edges of the Brillouin zone correspond usually to the minimum energy gap in a semiconductor. In graphene, this bandgap is zero. The carries behave in this case as relativistic particles (they behave as if they have no mass), or relativistic fermions. Characteristically for fermions, each energy state can be occupied only by one electron or a hole as well as they (the fermions) can be transported at ultrafast speed without back-scattering. The graphene crystal structure, the honeycomb symmetry results in very high optical phonon energy at the zone edges K and K` for transverse mode 37 . The energy values are ~ 198 meV at the zone centerand ~ 163 meV at the zone edges K and K` (Fig. 3.9).
Figure 3.9 Phonon band diagrams of graphene 37
The process of nonequilibrium energy relaxation in graphene takes place through carrier-phonon interaction through intra-valley, inter-valley, intra-band and inter-band transitions (Fig. 3.10).
Figure 3.10 Interaction of optical phonons and carriers through inter-valley, intra-valley, intra-band and inter-band transitions 37 .
Under optical pulse excitation, the intrinsic graphene has dominant carrier-carrier scattering that takes place in quasi equilibrium. Both, intra and inter-band optical phonons are taken into account but the inter-band Auger-like carrier-carrier scattering is neglected. The carrier distribution for the total energy and concentration of carriers is given in (3.7) 37 :
where ∑ and E are the carrier concentration and energy density, findex = quasi-Fermi distribution, τ(+/-) iO,inter and, τ(+/-) iO,intra are the inverse coefficients for the scattering rates for inter and intra-band optical phonons (where i = Γ for optical phonons close to the zone boundary with ωΓ = 198 meV and i = K for optical phonons close to the zone boundary with ωΓ = 163 meV), “+” stands for absorption and “-” for emission that are calculated by the Fermi golden rule, and υω is the Fermi velocity. The latter may be defined as following: even if we extract all possible energy from a Fermi gas by decreasing the temperature to absolute zero, the fermions can still move at a comparatively high speed. The fastest fermions move, in this case, at the velocity that corresponds to the Fermi energy, and this is the Fermi velocity. After the graphene is pumped with a photon energy of 0.8 eV, ɛF instantly falls down because of carrier coding and of the emission of optical phonons carrier at high-energy tails.
Further, ɛF becomes positive. When the pumping exceeds a certain threshold level, the population becomes inverted. Following the crossing of the threshold, the recombination process slows down (at ~ 10 ps). Although the population inversion is necessary condition it is not sufficient to achieve gain since we have the Drude absorption by carriers in graphene. It is the real part of the net dynamic conductivity Reσω (that consists of the intrabandand the interband that determines the gain). In particular, the negative values of Reσω produce the gain. Reσω is given by (3.8):
Reσω is proportional to the absorption of photons at frequency ω, where e = the electron charge;is the reduced Planck constant, kB is the Boltzmann constant, and τ is the momentum relaxation time of carriers. The value is always positive and is a loss contribution 37 .
The graphene conductivity presents substantial difficulties for its mathematical description which may be different at different frequencies in the THz range, in the upper THz range and up to far infrared, the expression derived from the Kubo formula 10 :
where the conduction band correspond to index 1 and the valence band to index 2,, and ; - the electron distribution function; and are the matrix elements of the velocity operator. It is assumed that the condition for the Fermi function are satisfied, i.e. The first part of Eq. (3.9) corresponds to the intra-band transitions and the second – to inter-band transitions.
The intra-band Drude conductivity is characteristic of the THz range. The frequency dependence of the conductivity is given in Fig. 3.11. The Fermi level at 100 meV is depicted by the dashed lines. The intra-band conductivity increases with an increase of doping. The solid lines represent this higher amount of doping at 200 meV. The intra-band Drude conductivity can be adjusted by doping and structural changes, the same way as it is done in usual semiconductor heterostructures.
Figure 3.11 Frequency dependence of the real part of the optical conductivity in graphene 38
Excitation of 2D plasmons in graphene causes extremely high plasmonic absorption and/or a giant plasmonic gain in the THz range. The gated plasmons are of particular significance for use in tunable graphene-based devices. The plasmons in gated devices have a super linear dispersion (Fig. 3.12).
Figure 3.12 Dispersion of plasmons in graphene in gated devices 38
In the semi-classical Boltzmann model describing the electron-hole plasma-wave dynamics in graphene, the plasmon phase velocity is proportional to the power of minus four of the gate bias and of the gate-to-graphene distance d (Fig. 3.13). The plasmons behavior in this case is quite different from those in conventional semiconductor quantum wells. The velocity always exceeds the Fermi velocity υF. The equal densities of electron and holes in graphene lead to two branches of carrier waves: the waves with neutral charge (sound-like waves) and the plasma waves. In the gates (heavily doped) both the majority and minority carriers exist. Subsequently, the minority carriers are damped and, consequently, only unipolar plasmons modes of the majority survive.
Figure 3.13 Plasma-wave velocity vs gate bias. The dashed line corresponds to the electron-hole sound-like waves measured in the vicinity of the neutrality point 38
An important feature is the possibility of tuning the plasmon frequencies over the whole THz range. There are several factors that are taken into account:
The presence of 2D plasmons in graphene substantially increases light-matter interaction improving the quantum efficiency 39 . If we pump graphene electrically or optically, we can achieve gain at THz frequencies and build lasers with higher power output than is available at the moment. In order to achieve this effect, there are two ways; one is propagation of the surface plasmon polaritons (SPPs) in graphene and the other is resonance of the SPPs 40 . In the former case, it is analytically revealed in Ref. 40 that when THz photons are incident with a TM mode to population-inverted graphene under pumping the THz photon could excite the SPPs so that they could propagate along with increasing the gain.
In such a situation, the absorption coefficient α is:
Figure 3.14 Frequency dependence of surface plasmon polaritons (SPP) absorption for monolayer-inverted graphene on SiO2/Si for quasi-Fermi energies 41
Figure 3.15 Frequency dependences of SPP gain for a monolayer of graphene 41
Regarding the latter way that exploits the SPP resonance, graphene is structured with dimensions on the order of micrometers or submictometers which is smaller than the wavelength of THz radiation. These structures acting as metamaterials, have plasmonic responses in the THz range 39 . Plasmons play a large role in the possibility to achieve negative conductivity or gain in graphene. Plasmons (in the classical interpretation) may be described as oscillations of free electron density. They are also a quantization of this kind of oscillation.
The incident THz radiation (Fig. 3.16) is amplified in graphene. Optical or electrical pumping inverts the carrier population. (In Fig. 3.16 an array of graphene microcavities has the parameters of L = 4 µm and a = 2 µm 42 ). The surface plasmons are excited by the incident THz photons with an effective field intensity vector component perpendicular to the direction of graphene ribbons with a – width of the ribbon and L – the period of graphene microcavities and stripes of metal. Surface plasmon polaritons (SPPs) produce significant gain at the frequencies that correspond to the SPP modes. An external THz wave incident on the planar array of the graphene microcavities is perpendicular to the substrate plane. The polarization of the electric field is directed across the metal strips. However, the stimulated emission of near IR (also possible range for the incident radiation) and of THz photons is limited by the quantum conductivity in the population-inverted graphene43 . The reason the limit imposed on absorbance by 2.3% is because the photons available for stimulated emission are produced only by the inter-band transition process 44 . In order to overcome the limit, shorter wavelength surface plasmon polaritons (SPP) were introduced. The resonant plasmon absorption in graphene materials from the SPP excitation is utilized to achieve superradiant THz emission. The pumping of the graphene is done either by optical illumination or by electron and hole injection from the p-and n-doped areas through the opposite sides of the metal stripes. The amplification is achieved by using a number of microcavities 45 . If the quasi Fermi energy level increases, the energy gain matches and then exceeds the net loss of energy due to the electron scattering in graphene 41 . In Fig. 3.17, a dependence of absorption as a function of the quasi-Fermi energy and the THz frequency for an array of the graphene microcavities is shown. The negative values of absorbance means that we have a gain.
Figure 3.16 a) Optical pumping of graphene 42 . b) A graphene-metal ribbon array. Silicon carbide may be used for the substrate 42
Figure 3.17 Contour map of the absorbance as a function of the quasi-Fermi energy and the frequency of the THz excitation radiation 42 .
The negative value of the absorbance gives the amplification coefficient. The valuecorresponds to the transparent graphene (when the loss equals gain). Also, at the transparency boundary, the THz wave amplification at the plasma resonance frequency becomes several orders of magnitude stronger. The behavior of the amplification coefficient close to the self-excitation regime is given in Fig. 3.18.
Figure 3.18 Variation of the power amplification coefficient 42
The plasmon oscillations are highly coherent and lasing takes place as soon as the gain at least equals the loss. The metal stripes in Fig. 3.16 act as synchronizing elements so that the plasmons in adjacent microcavities oscillate in phase. As a result, a single plasmon mode covering the whole array area, contributes to superradiant THz radiation 46 . The reported gain exceeds 104 cm−1 in the THz range 42 .
Compared to the current state-of-the-art THz sources, the graphene injection lasers 8 have relatively high quantum efficiency (~ 1), a higher output power (order of mW) and can operate at the room temperature, the advantages that are due to 1) the crystal structure of graphene that has no band gap and does not require any specific conditions for carrier depopulation which is not the case for the QCLs; 2) extremely long carrier relaxation times; 3) high optical phonon energy.
An intrinsic monolayer graphene sample was used for measurements at room temperature on a SiO2/Si substrate. A schematic description of the experiment is given in Fig. 3.19. The procedure is based on a time-resolved near-field reflective electro-optic sampling. The optimal pumping employs a fs – IR laser pulse. Simultaneously, a THz pulse is generated for probing the THz characteristics of the sample. As the emitter of THz probe pulse a 140 µm-thick CdTe crystal was used. An exfoliated monolayer-graphene on SiO2/Si substrate had an electro-optic sensor on its surface.
Figure 3.19 Cross-sectional image of the experimental set-up, THz probe, and optical probe 41
The optical pump and probe source was a femtosecond-pulsed laser with average power of about 4 mW with the duration of pulse of ~ 80 fs and repetition rate of 20 MHz. The laser beam used for optical pumping was mechanically chopped at ~ 1.2 kHz and linearly polarized. The chopping was necessary for the subsequent detection. The laser probing beam was cross-polarized with respect to the pumping beam’s polarization. In order to receive the envelope THz probe pulse, the CdTe layer was used. The primary pulse (#1 in Fig. 3.19) was detected from the primary THz beam at the top surface of CdTe layer. The THz pulse reflects from the interface at SiO2/Si after having penetrated inside and transmitted through the graphene. The pulse is electro-optically detected as a THz echo signal (#2 in Fig. 3.19). Thus, the temporal response consists of the first THz pulse that propagates through CdTe without interacting with the graphene and of a THz photon echo signal that actually probes the graphene. The delay between the above two pulses is equal to the roundtrip propagation time of the probe pulse through the layer of CdTe. The bandwidth of the THz response is restricted by the Restrahlen band of the CdTe crystal and is estimated to be approximately 6 THz. The observed gain spectra show qualitative coincidence with analytical calculations having threshold property against the pumping intensity 41 , 43 . The intensity of the observed THz echo pulse has a remarkable spatial dependence reflecting its polarization with a giant peak at the focused area where the THz probe pulse takes TM modes so that it can excite the SPPs in graphene. The observed gain enhancement effect is about 50 times as high as the other cases in which the SPPs could not be excited 41 .
A remarkable physical effect in graphene is a concentration of wave functions around classical periodic orbits or so-called “quantum scars”. In graphene whose dynamics, from the classical point of view, are chaotic, the semiclassical regime of a wave function can be regarded locally as a superposition of a number of plane waves. A nominal approach to the distribution of the wave functions states that the wave functions tend to concentrate on the paths analogous to unstable periodic orbits in classical systems. Bogomolny and Berry provided theoretical explanations based on the semiclassical Green’s function in nonlinear physics.
In addition to the existing works on nonrelativistic quantum mechanical systems where the dependence of the particle energy is quadratic, a new research described that scarring can occur also in relativistic quantum system described by the Dirac equation (as in the previous case, the Schrodinger equation is applicable) 47 . The latter phenomenon is important for device physics and device applications. In graphene, in particular (because of its hexagonal lattice structure), the band structure exhibits a linear dependence of the energy on the wave vector. The motion around the Dirac points is relativistic. The electron behavior in graphene resembles that of fermions which have the Fermi velocity υ = 106 cm/s 48 . The advantage lies in a potential capability of device made of graphene to operate at much higher speed than silicon devices can. Specifically, in quantum dots, at pointer states associated with scars, the conductance dependence exhibits narrower conductance resonant peaks. From the theoretical point of view, the relativistic quantum scars can be derived from the Dirac equation:
where σ is Pauli’s matrices andis a two component spinor describing the two types of nonequivalent atoms in graphene. On the whole, the scars in a relativistic quantum system have the same origin as the scars associated with ultrasonic fields, microwave and quantum dot billiards. Relativistic scars, however, are different from the conventional quantum scars.
In the presence of a magnetic field, the band structure of graphene changes which in its turn changes the scar pattern. In particular, it was found 47 that in a perpendicular weak magnetic field, the classical scar orbits remained almost unchanged, but in a strong field (with energies quantitized) the scars smear out. As a result, new orbits, both curved and straight, make appearance. The scar orbits could be more complicated, however, when spin effects take place.
Because of the relativistic nature of electron motion in graphene, a study of electronic transport is important for understanding quantum transport properties of the material from the fundamental and device design point of view 49 . In particular, the electron transport in quantum dots has a number of prominent features. The principal characteristics of quantum transport is conductance. Since the conductance is determined by transmission, it is sufficient to calculate the quantum transmission. With the change of electron energy, quantum dot transmission exhibits fluctuations. A correlation was found between the fluctuations and the appearance of scars inside the dots (such correlation may be found by investigating the local density of states (LDS)). L. Huang et al found that the LDS tends to concentrate in particular regions, around periodic orbits of quantum dots when such dots are treated classically. The local maxima or minima on the transmission curve indicate the above phenomenon, analogous to a one-dimensional finite square potential well. Similar to the finite square potential well, depending on the wave function phase, the resonances can either enhance or suppress transmission. Significantly, in conventional quantum dots in semiconductors, the conductance fluctuations are related to the presence of the scarring states.
For graphene quantum dots and nanoribbons, in particular, the standard Landauer formula applies. It relates the conductance G(EF) to the overall transmission TG(EF) 49 :
In order to calculate the transmission T(E) for graphene quantum dots, L. Huang et al used the low temperature conductance and non-equilibrium Green’s function (NEGF) formalism, as well as the division of the system into three parts (see Fig. 3.20).
Figure 3.20 a) Schematic representation of a quantum dot49 b) Tight-binding Hamiltonians for the Green’s function calculation. The left lead goes from -∞ to layer 0; the right lead – from layer 6 to ∞. N= 8, etc is the number of atoms in each layer 49 .