# $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
!--------------------------------------------------!