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The Unit Cell

The primitive GaN unit cell contains 4 atoms, in the case of the wurtzite structure (space group $\mathrm{P6_3mc}$), and 2 atoms, in the case of the zinc blende structure (space group $\mathrm{F\bar{4}3m}$). There are several equivalent ways to define the unit cells. For the purposes of these initial calculations we will define the structures as follows:

The shape of the wurtzite cell is a vertically oriented prism, with the base defined by the primitive lattice vectors, $\ensuremath{\mathbf{a}}$, and $\ensuremath{\mathbf{b}}$, which are of equal length and are separated by an angle of 60$^o$; $\ensuremath{\mathbf{a}}$ and $\ensuremath{\mathbf{b}}$ both lie in the horizontal $xy$-plane. The height of the cell is defined by the vector, $\ensuremath{\mathbf{c}}$, which is oriented vertically at 90$^o$ to both $\ensuremath{\mathbf{a}}$ and $\ensuremath{\mathbf{b}}$. In the ``ideal'' wurtzite structure $c$ is related to $a$ by $c=2\sqrt{\frac{2}{3}}a$; this is not necessarily the case in the real structure, as we will discuss in a moment.

To specify the positions of atoms within the cell we usually use fractional coordinates for convenience. If a point in space, $\ensuremath{\mathbf{r}}$, has Cartesian coordinates, $(x,y,z)$, then its fractional coordinates, $[x',y',z']$, are defined such that

\begin{displaymath}
\ensuremath{\mathbf{r}}=x'\ensuremath{\mathbf{a}}+y'\ensuremath{\mathbf{b}}+z'\ensuremath{\mathbf{c}}.
\end{displaymath} (2.1)

Note that we write fractional coordinates in square brackets to distinguish them from Cartesian coordinates.

The Ga atoms are positioned such that one is at the origin, $[0,0,0]$, and the other is at $[\frac{1}{3},\frac{1}{3},\frac{1}{2}]$. The N atoms are positioned directly above the Ga atoms. In the ``ideal'' wurtzite structure, these are at $[0,0,\frac{3}{8}]$ and $[\frac{1}{3},\frac{1}{3},\frac{7}{8}]$, so that the length of each Ga-N bond is the same if $c=2\sqrt{\frac{2}{3}}a$; a graphical representation of the ideal wurtzite cell is shown in Figure 2.1.

Figure 2.1: Primitive unit cell of wurtzite GaN. Ga atoms are represented by large grey spheres, and N atoms by smaller green spheres.
\includegraphics[scale=1.16,angle=0]{Figures/prim_WZ.eps}

Figure 2.2: 8-atom cubic cell of zinc blende GaN. Ga atoms are represented by large grey spheres, and N atoms by smaller green spheres.
\includegraphics[scale=1.16,angle=0]{Figures/bulk_ZB.eps}

However, in terms of cell symmetry, the vertical Ga-N bonds are not related to the diagonally oriented Ga-N bonds. Because of this, there is no a priory reason to expect these two sets of bonds to be the same length. There are therefore two extra degrees of freedom compared to the ideal structure - the length of the lattice vector, $\ensuremath{\mathbf{c}}$, relative to $\ensuremath{\mathbf{a}}$ and $\ensuremath{\mathbf{b}}$, and the vertical position of the N-atoms, relative to the Ga-atoms.

The deviation of the atomic coordinates from the ideal structure can be described in terms of a parameter, $d$, such that the positions of the N-atoms are given by $[0,0,\frac{3}{8}+d]$ and $[\frac{1}{3},\frac{1}{3},\frac{7}{8}+d]$.

The shape of the primitive 2-atom zinc blende cell is an equal-sided parallelepiped that can be most easily visualised with reference to a larger, 8-atom cubic cell, as shown graphically in Figure 2.2. This cubic cell has Ga-atoms at the origin and in the centre of each of the three faces that touch the origin. For each Ga-atom, there is a N-atom at a displacement of $[\frac{1}{4},\frac{1}{4},\frac{1}{4}]$ away from it. The lattice vectors defining the primitive cell are the three vectors going from the origin to the centre of the three faces where the Ga-atoms are. These vectors are of equal length and are separated from each other by angles of $60^o$. The three Ga-atoms on the faces of the cube are not in the primitive cell as they are simply the periodic repetitions of the atom at the origin. The primitive cell thus contains a Ga-atom at $[0,0,0]$ and a N-atom at $[\frac{1}{4},\frac{1}{4},\frac{1}{4}]$.

next up previous contents
Next: Choice of Pseudopotentials Up: Calculations on Bulk GaN Previous: Calculations on Bulk GaN   Contents
Stewart Clark 2012-08-09