**1. Introduction**

Graphene is a two-dimensional (2D) analogue of graphite (carbon, or C) material that has exceptional characteristics derived from the bonding characteristics of C bonding sheets. C has four valence electrons, with three of these electrons participating in σ-bonding with its closest neighbors, creating a honeycomb structure. [1] The fourth of these valence electrons occupies an orbital perpendicular to the one-dimensional (1D) sheet creating delocalized π-bonding, as shown in Figure 1, which allows for the creation of a two-dimensional electron gas (2DEG) with high mobility within the sheets. [1, 2]

**Figure 1.** Graphene geometry, bonding, and a related band diagram [1].

The delocalization of the π-bonding electrons allows for the graphene sheets to have high mobility, up to 15,000–200,000 cm2 /Vs, limited by interactions with the substrate, any contam‐ inant particles, or from itself during bilayer growth. [1, 3–7] This makes cleanliness, grain size, and substrate interference very important issues for growing and using graphene for high mobility and ultrafast applications.

In this review, we are going to focus on the important electrical properties of graphene; however, we should mention some of its other properties for completeness. Due to the 2D nature of a graphene crystal, a single flake will exhibit a large breaking strength of ≈40 N/m due to the absence of slip planes associating the fracture strength of graphene with the strong bonding of c–c in a hex ring. [8] The isolation of electrons from phonons also contributes to the high room temperature thermal conductivity of ≈5,000 W/mK. [9] Along with its high breaking strength graphene is also very pliable with a Young's modulus ≈1.0 TPa and an elastic strain of up to 20%. [8] These values were partially expected on the basis of previous studies of carbon nanotubes and graphite; although the higher values observed in graphene can be attributed to the crystal defects in samples obtained by micromechanical cleavage.

There are even more intriguing material characteristics of graphene such as shrinkage with increasing *T* at all *T* due to membrane phonons dominating in 2D. [10] Also, graphene exhibits simultaneously high pliability (folds and pleats are commonly observed) and brittleness (it fractures like glass at high strains). [11] Equally unprecedented is the observation that the oneatom-thick film is impermeable to gases, including helium. [12]

For electronic applications the structure of graphene creates a semi-metal with a direct Fermi-Dirac band structure, as shown in Figure 1, having charge carriers interacting as Dirac Fermions (with zero-effective mass) that allows for ballistic transport of up to a micron at room temperature. [13– 15] The Fermi-Dirac cone as shown in Figure 1c, however, is modified either by the addition of multiple layers as shown in Figure 2iii, the addition of two layers and doping from contaminant particles (metal or polymer particles lying on the surface) shown in 2iv, or contaminants doping a single layer as shown in Figure 2ii. [16] The contaminant-induced doping would move the Fermi level either up or down, the Dirac cone causing a rounding of the k states resulting in a decrease in the mobility of the current carriers (electron or holes). [16] This, along with the thickness restriction for graphene, creates large resistance and chemical inertness, unless chemically doped and functionalized, making its use for pure conductive applications less attractive. [16, 17]

**1. Introduction**

60 Graphene - New Trends and Developments

with high mobility within the sheets. [1, 2]

**Figure 1.** Graphene geometry, bonding, and a related band diagram [1].

mobility, up to 15,000–200,000 cm2

mobility and ultrafast applications.

Graphene is a two-dimensional (2D) analogue of graphite (carbon, or C) material that has exceptional characteristics derived from the bonding characteristics of C bonding sheets. C has four valence electrons, with three of these electrons participating in σ-bonding with its closest neighbors, creating a honeycomb structure. [1] The fourth of these valence electrons occupies an orbital perpendicular to the one-dimensional (1D) sheet creating delocalized π-bonding, as shown in Figure 1, which allows for the creation of a two-dimensional electron gas (2DEG)

The delocalization of the π-bonding electrons allows for the graphene sheets to have high

inant particles, or from itself during bilayer growth. [1, 3–7] This makes cleanliness, grain size, and substrate interference very important issues for growing and using graphene for high

In this review, we are going to focus on the important electrical properties of graphene; however, we should mention some of its other properties for completeness. Due to the 2D nature of a graphene crystal, a single flake will exhibit a large breaking strength of ≈40 N/m due to the absence of slip planes associating the fracture strength of graphene with the strong bonding of c–c in a hex ring. [8] The isolation of electrons from phonons also contributes to the high room temperature thermal conductivity of ≈5,000 W/mK. [9] Along with its high breaking strength graphene is also very pliable with a Young's modulus ≈1.0 TPa and an elastic strain of up to 20%. [8] These values were partially expected on the basis of previous studies of carbon nanotubes and graphite; although the higher values observed in graphene can be attributed

There are even more intriguing material characteristics of graphene such as shrinkage with increasing *T* at all *T* due to membrane phonons dominating in 2D. [10] Also, graphene exhibits simultaneously high pliability (folds and pleats are commonly observed) and brittleness (it fractures like glass at high strains). [11] Equally unprecedented is the observation that the one-

to the crystal defects in samples obtained by micromechanical cleavage.

atom-thick film is impermeable to gases, including helium. [12]

/Vs, limited by interactions with the substrate, any contam‐

**Figure 2.** (i) Diagram showing the Dirac Fermi cone; (ii) the modification of the k states by chemical or geometry re‐ strictive doping; (iii) the modification of the k states by bilayer graphene; (iv) and finally, the modification of the k states in doped bilayer graphene. [16]

For applications such as the channel in a field effect transistor, graphene provides an interest‐ ing solution since it can be doped electrostatically and has extremely high mobility allowing for quick response. [18] The replacement of Si by graphene for logic gates might be considered due to the high potential switching speed; however, the absence of a band gap means that a relatively large band gap would have to be induced through a variety of doping or other symmetry breaking mechanisms. [18] The absence of a band gap in graphene limits voltage and power gains that is achieved through operation of a device in the saturation regime, along with having a low Ion/Ioff ratio. [16] To overcome this, several doping strategies as shown in Figure 3 have been proposed and tested including: electrostatic doping, chemical doping, and stress or geometry restricted doping by breaking the graphene periodicity (and band proper‐ ties). [18]

The induction of a band gap has been attempted by multiple groups creating transistors with low on/off ratio and high mobility with a tradeoff between on/off ratio and mobility possible through graphene functionalization techniques.[1] This makes graphene more desirable for

**Figure 3.** Diagram showing multiple mechanisms for inducing a band gap in graphene [19].

applications that require fast response times, but not necessarily big on/off ratios such as RF electronics and IR detectors.
