# Electronic and Transport Properties of Epitaxial Graphene on SiC and 3C-SiC/Si: A Review
## 2 Electronic Band Structure  
### Monolayer Graphene  
The electronic properties of graphene were first described by [PERSON] in 1947 [22] who used the nearest neighbour tight-binding model (involving only p\({}_{x}\) electrons perpendicular to the plane of graphene, i.e., p\({}_{x}\) orbitals--which result in \(\pi\) bands) to approximate the low energy electronic structure of an infinite graphene lattice, which demonstrates the linear dispersion.  
\[\mathbf{E}(\mathbf{k})=\pm\tau\sqrt{1+4\mathbf{cos}\bigg{(}\frac{\sqrt{3}}{2}\mathbf{ak}_{x} \bigg{)}\mathbf{cos}\bigg{(}\frac{1}{2}\mathbf{ak}_{y}\bigg{)}+4\mathbf{cos}^{2}\bigg{(} \frac{1}{2}\mathbf{ak}_{y}\bigg{)}} \tag{1}\]  
The linear dispersion relation of electrons in single-layer graphene can be written using the strong-coupling approximation by taking into account only the nearest-neighbor interaction (low-energy approximation) as follows.  
\[\mathbf{E}(\mathbf{\Delta}\mathbf{k})=\mathbf{cr}\mathbf{h}\mathbf{v}_{\mathbf{F}}\ \mathbf{\Delta}\mathbf{k} \tag{2}\]  
where \(\mathbf{cr}\mathbf{h}\) is the reduced Planck's constant, the proportionality or slope is the Fermi velocity, \(\mathbf{v}_{\mathbf{F}}\) is approximately equal to \(10^{6}\) m s\({}^{-1}\), and \(\mathbf{\Delta}\mathbf{k}\) is the momentum relative to the \(\mathbf{K}\) points of the hexagonal reciprocal unit cell [23]. The band structure of a monolayer graphene (MLG), shown in Figure 1, exhibits the gapless linear dispersion (in k-space known as the Dirac cone) of the \(\pi\) bands at the \(\mathbf{K}\)-point. This has two important implications. The first is the charge carriers, Dirac fermions, which are massless. The second is semi-metallic behaviour [24; 22] because when the Fermi energy, \(\mathbf{E}_{\mathbf{F}}\), is located where the dispersion converges to a point (the Dirac point, \(\mathbf{E}_{\mathbf{D}}\)), the density of states is zero, and electronic conduction is possible only through thermally excited electrons [25]. The linear dispersion of monolayer graphene results in several extraordinary electronic properties such as an anomalous quantum Hall effect with Berry's phase of \(\pi\)[8; 26; 27; 28; 29].  
The tight-binding model oversimplifies the situation of EG on SiC(0001) as a buffer layer exists between SiC and EG. [PERSON] [30] and [PERSON] et al. [31] performed a first-principles calculation, which included the buffer layer and determined the \(\mathbf{E_{\mathrm{F}}}\) was about 450 meV above the \(\mathbf{E_{\mathrm{D}}}\) due to doping of the graphene layer by charge transfer from the buffer layer. This is in agreement with angle-resolved photoelectron spectroscopy (ARPES) measurements [26; 32; 33; 34]. [PERSON] et al. [35] found the average Fermi velocity, obtained from the slope of \(\mathbf{E(\Delta k)}\) as \(\mathbf{v_{\mathrm{F}}}=1.0\ (\pm 0.05)\times 10^{6}\ \mathrm{ms^{-1}}\) for energies down to \(\sim\)500 meV above \(\mathbf{E_{\mathrm{D}}}\)[19; 36; 37]. This value of \(\mathbf{v_{\mathrm{F}}}\) is larger than the \(\mathbf{v_{\mathrm{F}}}\) for bulk graphite (\(\mathbf{v_{\mathrm{F}}}=0.86\times 10^{6}\ \mathrm{ms^{-1}}\)) [38].  
[PERSON] et al. measured the band structure of the monolayer EG/3C-SiC(111) using ARPES at room temperature [19] (see Figure 2) and found the \(\mathbf{E_{\mathrm{F}}}\). Using the linear dispersion, the charge carrier concentration, \(\mathbf{n}\), of monolayer graphene can be obtained as:  
\[\mathbf{n}=(\mathbf{E_{\mathrm{F}}}-\mathbf{E_{\mathrm{D}}})^{2}/(\pi)(\mathbf{t}\mathbf{v_{ \mathrm{F}}})^{2} \tag{3}\]  
where \(\mathbf{h}\) is the [PERSON]'s constant, \(\mathbf{v_{\mathrm{F}}}=1.1\times 10^{6}\ \mathrm{m}\ \mathrm{s^{-1}}\), and \(\mathbf{E_{\mathrm{F}}}\) is 500 meV. This resulted in \(\mathbf{n}\sim 2\times 10^{13}\ \mathrm{cm^{-2}}\).  
The influence of graphene-substrate interaction on \(\mathbf{v_{\mathrm{F}}}\) of monolayer epitaxial graphene was estimated by [PERSON] [39] in which the substrate interaction reduces the \(\mathbf{v_{\mathrm{F}}}\) of electrons.