6.1 Geometric Properties of the First Brillouin Zone

The crystal structure of silicon is known as diamond structure which is adopted by solids with four symmetrically placed covalent bonds. The diamond structure can be described by a face-centered cubic (FCC) lattice with a basis of two atoms where one is placed at $a_0(0,0,0)^T$ and the other at $a_0($¼$,$¼$,$¼$)^T$ , where $a_0$ is the lattice constant which is about $a_0=0.543 \MR{nm}$ for relaxed silicon. Figure 6.1 shows the conventional and the primitive cell of the diamond structure, where Figure 6.1(a) shows in addition the basis vectors of the direct lattice, cf. Equation (6.2).

Figure 6.1: Silicon lattice, known as diamond structure is adopted by solids with four symmetrically placed covalent bonds. The translational symmetry is a FCC lattice with a basis of two atoms, one at $a_0(0,0,0)^T$ an the other at $a_0($¼$,$¼$,$¼$)^T$ , where $a_0$ is the lattice constant.

The lattice structure is invariant under translation in the real space of the form

$$\vec{r} '=\vec{r}+ i\vec{a}_1 + j\vec{a}_2 + k\vec{a}_3,$$ (6.1)

where $i$ ,$j$ and $k$ are integers. The basis vectors of the direct lattice are

$$\vec{a}_1 = \frac{a_0}{2}\begin{pmatrix}0 \ 1 \ 1 \end{pmatrix}, \qquad \vec{a}_2 = \frac{a_0}{2}\begin{pmatrix}1 \ 0 \ 1 \end{pmatrix}, \quad \vec{a}_3 = \frac{a_0}{2}\begin{pmatrix}1 \ 1 \ 0 \end{pmatrix}$$ (6.2)

where $a_0$ denotes the lattice constant, see Figure 6.1(a).

The set of reciprocal primitive vectors $(\vec{b}_1,\vec{b}_2,\vec{b}_3)$ of the lattice vectors $(\vec{a}_1,\vec{a}_2,\vec{a}_3)$ are determined by using matrix inversion of a column vector representation which reads

$$\begin{pmatrix}b_{1x} & b_{2x} & b_{3x} \ b_{1y} & b_{2y} & b_{3... ...3x} \ a_{1y} & a_{2y} & a_{3y} \ a_{1z} & a_{2z} & a_{3z} \end{pmatrix}^{-1}.$$ (6.3)

The respective general reciprocal lattice vector in $\mathbf{k}$ -space is given by

$$\vec{G}_{i,j,k}=i\vec{b}_1 + j\vec{b}_2 + k\vec{b}_3,$$ (6.4)

where $i$ ,$j$ and $k$ are integers. The basis vectors of the reciprocal lattice are

$$\vec{b}_1 = \frac{2\pi}{a_0}\begin{pmatrix}-1 \ 1 \ 1 \end{pmat... ... , \qquad \vec{b}_3 = \frac{2\pi}{a_0}\begin{pmatrix}1 \ 1 \ -1 \end{pmatrix}$$ (6.5)

with $\vec{a}_i ^T \cdot \vec{b}_j = 2 \pi \delta_{i,j}$ . The Wigner-Seitz cell [113] of the reciprocal lattice, referred as the first Brillouin zone, is bordered by $14$ faces which can be given as

$$\vert k_x \vert + \vert k_y \vert + \vert k_z \vert = \frac{3}{2}... ...d \vert k_x \vert = \frac{2\pi}{a_0}, \quad \vert k_y \vert = \frac{2\pi}{a_0}, \quad \vert k_z \vert = \frac{2\pi}{a_0},$$ (6.6)

as depicted in Figure 6.2.

Figure 6.2: The reciprocal lattice structure of a face-centered cubic (FCC) basis forms a body-centered cubic (BCC) lattice. Figure 6.2(a) shows a primitive reciprocal lattice part and the Wigner-Seitz cell which is referred as the first Brillouin zone. Figure 6.2(b) shows the periodicity of the Brillouin zone cells.