Next: Choice of Pseudopotentials
Up: Calculations on Bulk GaN
Previous: Calculations on Bulk GaN
Contents
The Unit Cell
The primitive GaN unit cell contains 4 atoms, in the case of the wurtzite structure
(space group
),
and 2 atoms, in the case of the zinc blende structure (space group
).
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,
, and
, which are of equal length and are separated by
an angle of 60;
and
both lie in the horizontal plane.
The height of the cell is defined by the vector,
, which is oriented
vertically at 90 to both
and
. In the ``ideal'' wurtzite structure
is related to by
; 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,
, has Cartesian coordinates, , then its
fractional coordinates, , are defined such that

(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, , and the other
is at
.
The N atoms are positioned directly
above the Ga atoms. In the ``ideal'' wurtzite structure, these are
at
and
, so that the
length of each GaN bond is the same if
; 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.

Figure 2.2:
8atom cubic cell of zinc blende GaN. Ga atoms are represented
by large grey spheres, and N atoms by smaller green spheres.

However, in terms of cell symmetry, the vertical GaN bonds are not related to the
diagonally oriented GaN 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,
, relative to
and
, and the vertical position
of the Natoms, relative to the Gaatoms.
The deviation of the atomic coordinates from the ideal structure can be described
in terms of a parameter, ,
such that the positions of the Natoms are given by
and
.
The shape of the primitive 2atom zinc blende cell is an equalsided parallelepiped that
can be most easily visualised with reference to a larger, 8atom cubic cell,
as shown graphically in Figure 2.2.
This cubic
cell has Gaatoms at the origin and in the centre of each of the three faces that touch
the origin. For each Gaatom, there is a Natom at a displacement of
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 Gaatoms are. These vectors are of equal length and are separated
from each other by angles of . The three Gaatoms 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 Gaatom at and a Natom at
.
Next: Choice of Pseudopotentials
Up: Calculations on Bulk GaN
Previous: Calculations on Bulk GaN
Contents
Stewart Clark
20120809