1.5 Graphene- and Silicon-based Nanostructures

Bismuth and its compounds that are commonly used in thermoelectric applications [5] suffer from high cost. In addition, tellurium compounds cannot be used in large-scale applications due to rarity. Silicon, on the other hand, is the second most abundant element on earth and has been used in large scale manufacturing processes. Silicon is the most widely used material in semiconductor industry and its fabrication process are optimized. Silicon crystallizes in a diamond structure with lattice constant of $a_c \sim 0.54~\mathrm{nm}$ (see Fig. 1.9) and has a bandgap of nearly $1.1~\mathrm{eV}$ and thermal conductivity of $\kappa_l=149~\mathrm{W/mK}$ at room temperature. Due to high thermal conductivity at room temperature bulk silicon has $ZT\approx 0.01$ , which makes it a very poor thermoelectric. However, recent experimental studies showed that the thermal conductivity is sharply reduced in silicon-based nanostructures, i.e. nanowires and thin layers [17,20,21]. The large reduction in the thermal conductivity was attributed to enhanced scattering of phonons on the surfaces of the nanochannels. As a result, $ZT$ values of about $0.6$ were achieved at room temperature, a large improvement compared to $ZT$ of bulk silicon. Furthermore, it should be possible to reach even higher values of $ZT$ using techniques for optimizing the geometry, transport and confinement orientations, and confinement size.

Figure 1.9: Diamond crystal structure of silicon.

On the other hand, graphene, a recently discovered form of carbon, has received much attention over the past few years due to its excellent electrical, optical, and thermal properties [22]. As shown in Fig. 1.10, carbon atoms in graphene are tightly packed into a two-dimensional (2D) honeycomb lattice due to their $\mathrm{sp^2}$ hybridization. The primitive unit cell is defined by two lattice vectors $\vec{a}_1$ and $\vec{a}_2$ :

$$\vec{a}_1=\frac{3}{2}a_{cc}\hat{x}+\frac{\sqrt{3}}{2}a_{cc}\hat{y}$$ (1.18)

and

$$\vec{a}_1=\frac{3}{2}a_{cc}\hat{x}-\frac{\sqrt{3}}{2}a_{cc}\hat{y}$$ (1.19)

where $a_{cc}=0.14~\mathrm{nm}$ is carbon-carbon bond length. The lattice (unit cell) is composed of two sublattices called $A$ and $B$ . Although the electrical conductance of graphene is as high as that of copper [23], as a zero bandgap material, pristine graphene has a small Seebeck coefficient [24]. However, one can open up bandgaps by appropriate patterning of graphene sheets [25,26,27]. Many theoretical studies have been recently performed on the thermal conductivity of graphene-based structures as well. It has been shown that boundaries and edge roughness can strongly degrade [19] its high thermal conductance [28,29]. Recently, a large scale method to produce graphene sheets has been reported [30], which could pave the way for large scale graphene applications. These factors render graphene as a candidate for future thermoelectric applications.

Figure 1.10: The crystal structure for graphene is defined by lattice vectors $\vec{a}_1$ and $\vec{a}_2$ , and the basis includes two carbon atoms called type $A$ and type $B$ .