$wz-restrictions¶
Some restrictions apply for wurtzite materials.
$wz-restrictions required miller-size integer required miller-default-direction-of-x integer_array required miller-default-direction-of-y integer_array required direction-cosines double_array required miller-direction-of-cx integer_array required miller-direction-of-cy integer_array required miller-direction-of-cz integer_array required lattice-constants-for-cxyz double_array required $end_wz-restrictions required
Explanations
- miller-size
- type:
integer
- presence:
required
- value:
4
There are four Miller-Bravais indices altogether that define the (hkil) plane.
Note
They do not define the [hkil] direction.
Usually for wurtzite, the four-digit Miller-Bravais indices (h k i l) are used.
- miller-default-direction-of-x
- type:
integer_array
- presence:
required
Four-digit Miller indices of the (hkil) plane perpendicular to x axis of simulation coordinate system.
- miller-default-direction-of-y
- type:
integer_array
- presence:
required
Four-digit Miller indices of the (hkil) plane perpendicular to y axis of simulation coordinate system.
miller-default-direction-of-x = 1 0 -1 0 ! ( 1 0 -1 0) plane miller-default-direction-of-y = -1 2 -1 0 ! (-1 2 -1 0) plane
This corresponds to x axis and y axis, respectively, in simulation coordinate system, i.e. the x axis is perpendicular to the ( 1 0 -1 0) plane and the y axis is perpendicular to the (-1 2 -1 0) plane, respectively.
These value can be overwritten in $domain-coordinates (hkil-x-direction, hkil-y-direction, hkil-z-direction).
Note
It holds for the four-digit Miller-Bravais indices (h k i l): i = - (h + k), i.e. i is not independent.
Direction cosines
- direction-cosines
- type:
double_array
- presence:
required
- value:
-0.5 0.0 0.0
Direction cosines between lattice vectors. g1*g2, g2*g3, g1*g3 gi … unit vectors in lattice directions.
Direction cosine refers to the cosine of the angle between any two vectors. Direction cosines are useful for forming direction cosine matrices that express one set of orthonormal basis vectors in terms of another set, or for expressing a known vector in a different basis. For wurtzite, we use:
( 1 -0.5 0 ) g_ik = ( -0.5 1 0 ) ( 0 0 1 )
Four-digit Miller indices of the (hkil) plane:
- miller-direction-of-cx
- type:
integer_array
- value:
1 0 -1 0
Corresponds to x axis in cartesian crystal coordinate system, i.e. the x axis is perpendicular to the ( 1 0 -1 0) plane.
- miller-direction-of-cy
- type:
integer_array
- value:
-1 2 -1 0
Corresponds to y axis in cartesian crystal coordinate system, i.e. the y axis is perpendicular to the (-1 2 -1 0) plane.
- miller-direction-of-cz
- type:
integer_array
- value:
0 0 0 1
Corresponds to z axis in cartesian crystal coordinate system, i.e. the z axis is perpendicular to the (0 0 0 1) plane, i.e. axis parallel to sixfold rotational axis in wurtzite which is coincidently also the [0001] direction.
miller-direction-of-cx = 1 0 -1 0 ! (10-10) plane miller-direction-of-cy = -1 2 -1 0 ! (-12-10) plane miller-direction-of-cz = 0 0 0 1 ! (0001) plane ("[0001] direction")
These are the default orientations.
- lattice-constants-for-cxyz
- type:
double_array
- value:
1.0 1.0 1.6329931618554520654648560498039
Lattice constants to interpret the Miller-Bravais indices: \(1.0\), \(1.0\), \(\sqrt{8/3}\). Here, we take the ideal wurtzite ratio of \(c/a=\sqrt{8/3}\).
In wurtzite, there are three coordinate axis in the basal plane, \({\mathbf{a}}_1\), \({\mathbf{a}}_2\), \({\mathbf{a}}_3\), and the \({\mathbf{c}}\) direction perpendicular to it. There are different definitions for it.
\({\mathbf{a}}_1=\) [10-10], \({\mathbf{a}}_2=\) [-12-10], \({\mathbf{a}}_3=\) […], \({\mathbf{c}}=\) [0001] (used by nextnano³) ==> \({\mathbf{a}}_1=\sqrt{3}/2 a {\mathbf{x}} - a/2 {\mathbf{y}}\), \({\mathbf{a}}_2=a {\mathbf{y}}\), \({\mathbf{c}}=c {\mathbf{z}}\)
\({\mathbf{a}}_1=\) [2-1-10], \({\mathbf{a}}_2=\) [-12-10], \({\mathbf{a}}_3=\) [-1-120], \({\mathbf{c}}=\) [0001] ==> \({\mathbf{a}}_1=\) [2-1-10] \(a/\sqrt{6}\), \({\mathbf{a}}_2=\) [-12-10] \(a/\sqrt{6}\), \({\mathbf{c}}=\) [0001] \(c=\) [0, 0, 0, 3 \(\lambda\) ] \(a/\sqrt{6}\), where \(\lambda = \sqrt{2/3} c/a\).
Example
!--------------------------------------------------! $wz-restrictions miller-size = 4 miller-default-direction-of-x = 1 0 -1 0 miller-default-direction-of-y = -1 2 -1 0 direction-cosines = -0.5 0.0 0.0 ! g1*g2, g2*g3, g1*g3 gi ... unit vectors in lattice directions miller-direction-of-cx = 1 0 -1 0 miller-direction-of-cy = -1 2 -1 0 miller-direction-of-cz = 0 0 0 1 lattice-constants-for-cxyz = 1.0 1.0 1.6329931618554520654648560498039 $end_wz-restrictions !--------------------------------------------------!