2.1.1 Electronic Structure

Figure 2.1: Two dimensional rectangular structure.

The empirical tight-binding model is a standard, convenient, and accurate method for calculating the electronic structure of semiconductors. It is also referred to as Bloch or linear combination of atomic orbitals (LCAO) method [31]. The Hamiltonian matrix for a simplified 2D system, using the TB method is described as follows: The 2D structure in Fig. 2.1 is composed of $n_y$ chains of lattice points (i.e. atomic sites), each chain with $n_x$ sides. Assuming that each point is represented by one basis orbital, the Hamiltonian matrix will have the size  $(n_{x}n_{y})\times(n_{x}n_{y})$ , and is given by:

$$H = \left ( \begin{array}{cccccc} \alpha & \beta & 0 & 0 & 0 & ..... ... 0 & 0 & 0 & \beta^{\dagger} & \alpha \end{array} \right)_{n_xn_y\times n_xn_y}$$ (2.2)

Figure 2.2: Schematic representation of the nearest neighbors of the $i^{th}$ carbon atom. Up to four nearest-neighbors are included.

where

$$\alpha = \left ( \begin{array}{cccccc} 2(t_x+t_y) & -t_y & 0 & 0 ... ...dots \\ ... & 0 & 0 & 0 & -t_y & 2(t_x+t_y) \end{array} \right)_{n_y\times n_y}$$ (2.3)

and

$$\beta = \left ( \begin{array}{ccccc} -t_x & 0 & 0 & 0 & ...\\ 0 &... ... \vdots & \vdots \\ ... & ...& 0 & 0 & -t_x \end{array} \right)_{n_y\times n_y}$$ (2.4)

where the $[\alpha]_{n_{y}\times n_{y}}$ submatrix describes the coupling within each chain, and the $[\beta]_{n_{y}\times n_{y}}$ submatrix describes the coupling between the adjacent chains. In general, the size of these components would be $n_b \times n_b$ , where $n_b$ is the number of basis orbitals in each unit cell. The eigenvalues of $H$ are the corresponding energies for the electrons in the structure and can be adjusted by fitting the parameters $t_x$ and $t_y$ . In the case of an open system, the boundary conditions are implemented by modifying the Hamiltonian to:

$$H = \left ( \begin{array}{cccccc} \alpha & \beta & 0 & ... & 0 & ... ... & ... & 0 & \beta^{\dagger} & \alpha \end{array} \right)_{n_xn_y\times n_xn_y}$$ (2.5)

The only change, here, is in the submatrices $H(1, n_{x})$ and $H(n_{x}, 1)$ , which now include the self energies of the left and right contacts. In the case of periodic boundary conditions, the bandstructure is calculated by considering the unit cell (index $n$ ) of the lattice, connected to the neighboring unit cells (index $m$ ) using the matrix elements $[H_{nm}]$ . For example, as indicated in the structure of Fig. 2.1, once periodic boundary conditions are applied along the $x$ -axis, then $H_{nn}=\alpha$ and $H_{n,n+1}=\beta$ for all $n \in [1,n_x]$ . The bandstructure of the lattice is then obtained by calculating the eigenvalues of the Hamiltonian as:

$$[h(k)]=\sum_{m}H_{nm}\mathrm{e}^{i\vec{k}\cdot(\vec{d}_m-\vec{d}_n)}$$ (2.6)

for each $\vec{k}$ -point in the Brillouin zone (BZ) [32]. Here, $\vec{d}_m$ is the vector corresponding to the position of neighboring unit cell $m$ and the sum is over all neighboring unit cells.

Figure 2.3: Electronic band structure of Graphene along the high-symmetry band line is calculated employing a third nearest tight binding approximation. Experimental results are taken from Ref. [33]

In the case of graphene, a third nearest neighbor tight-binding model is used to describe its electronic structure. In this case, the particular atom $i$ and its nearest neighbor atoms are shown in Fig. 2.2. The hopping parameter between two nearest atoms separated by distance $a_{cc}$ is $t_1=-3.2~\mathrm{eV}$ . The tight-binding parameter of the third-nearest neighbor atoms located $2a_{cc}$ away from each other is $t_3=-0.3~\mathrm{eV}$  [34]. The hopping parameter for the second nearest-neighbor is assumed to be $0~\mathrm{eV}$ . The bandstructure of graphene along the high-symmetry band line is shown in Fig. 2.3. The tight-binding results are in good agreement with experimental data taken from [33], in particular around the Fermi energy $E_{\mathrm{F}}=0$ which dominates the electrical properties. As shown in Ref. [34] this method can capture the details of the bandstructure of graphene-based nanostructures. The tight-binding model with calibrated parameters provides band-gap and subband-edge energies in excellent agreement with first-principles calculations [34].