**3.3. Geometry restriction**

**Figure 13.** Image showing that the misalignment of 2D materials can electrically isolate the two sheets by separating

**Figure 12.** Diagram showing the stacking of multiple Van der Waals materials in order to create unique and tunable

It has been shown that a twist angle between two graphene sheets above 2° electrically isolates the two graphene sheets from one another except at certain twist angles as shown in Figure 13. [49] Most 2D materials have also been shown to have intrinsic doping due to vacancies and edge defects that create more problems for device integration. [49] It should be noted that the mobilities in graphene on boron nitride (BN) substrates have been measured up to 140,000

/Vs, which is very close to completely suspended graphene grown from a SiC step edge, showing the validity of using 2D heterostructures for device integration and isolation. [32, 42]

the Dirac cones [50].

electrical properties [47]

70 Graphene - New Trends and Developments

cm2

The final way to dope graphene is by breaking the lattice periodicity of graphene as shown in Figure 15. [19, 53] This can be done by reducing the size of a graphene sheet in one direction so that the Fermi levels from the periodic boundary conditions are refined, providing doping through a quantum confinement effect. [53] Quantum confinement occurs when the material dimensions are below the Bohr radius, which for graphene is at 10 nm. [13, 53] This has been shown to be accomplished through the patterning of graphene into ribbons with one dimen‐ sion restricted to under 10 nm, thus opening a gap of 2.5–3.0 eV in theory and 0.5 eV experi‐ mentally. [19, 53] Graphene with a size in either x or y under 10 nm is known as a graphene nanoribbon and it suffers, like many other graphene synthesis techniques, from a lack of a reliable production technique. [53] Traditional semiconductor line definition techniques cannot reliably get a line definition below 20 nm, with large problems creating lines with acceptable line edge roughness. For graphene nanoribbons, the resistance induced through scattering from the line edge roughness is coupled with a lack of graphene conformality, not knowing whether the line definition will create "zig-zag" or "arm-chair" end terminations that provide different conductivity values. [19, 53] The difference between "zig-zag" and "armchair" end terminations is shown in Figure 16 and the difference in conductivities between the two create a discrepancy when designing a device utilizing multiple nanoribbons or multiple devices utilizing a graphene nanoribbon. [54]

**Figure 15.** Different defined graphene sheet edge states and the associated band diagrams showing opening according to edge definitions [52].

**Figure 16.** A picture showing the difference between zig-zag and armchair graphene end terminations [54].

The induced line edge roughness produces many scattering defect reducing the lattice periodicity, obliterating the induced band gap, and decreasing the mobility ultimately limiting the usefulness of graphene nanoribbon formation. [16, 19] Thus, to achieve useful devices from geometry restricted graphene, a reliable method of patterning graphene with low line edge roughness and uniform width must be developed.
