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Domain coordinates
Orientation of crystal relative to simulation coordinate
system
The overall simulation domain, that is the real space region in which the
device is defined, is restricted to a cuboid, a rectangle, or a line - dependent
on the dimension for the simulation. The extension of this domain must be
specified by its minimum and maximum value along the corresponding
coordinate axes. The complete simulation domain will be "filled" with
material information later.
Additionally, the orientation of the global three-dimensional coordinate
system must be oriented with respect to the crystal and some information about a
substrate for pseudomorphic growth must be given.
Keyword - Specifier syntax
!----------------------------------------------------!
$domain-coordinates required ! The global simulation domain is
restricted to be a cuboid, a plane or a line.
domain-type
integer_array
required ! 1 1 1
for xyz-grid, 1 0 1
for xz-grid, 1 1 0
for xy-grid ..., 1 0 0
for x-grid
! redundant
information: $simulation-dimension, orientation = ...
is now used. However, this specifier must be still there (CHECK: WHY???).
x-coordinates
double_array
optional ! xmin,xmax of simulation domain
y-coordinates double_array
optional ! ymin,ymax of simulation domain
z-coordinates
double_array
optional ! zmin,zmax of simulation domain
!
hkl-x-direction-zb
integer_array optional ! Miller
indices of plane perpendicular to x-coordinate
axis of simulation coordinate system, e.g. (1 0 0) which
corresponds to [100] direction
hkl-y-direction-zb
integer_array optional ! y-coordinate
axis of simulation coordinate system, e.g. (0 1 0) which
corresponds to [010] direction
hkl-z-direction-zb
integer_array optional ! Miller
indices of plane perpendicular to z-coordinate
axis of simulation coordinate system, e.g. (0 0 1) which
corresponds to [001] direction
!
hkil-x-direction
integer_array optional !
four-digit Miller-Bravais indices of (hkil) plane that is
perpendicular to x-coordinate
axis direction
hkil-y-direction
integer_array optional ! four-digit Miller-Bravais
indices of (hkil) plane that is perpendicular to y-coordinate
axis direction
hkil-z-direction
integer_array optional !
four-digit Miller-Bravais indices of (hkil) plane that is
perpendicular to z-coordinate
axis direction
growth-coordinate-axis integer_array optional !
pseudomorphic-on character optional !
a default material or user defined name
alloy-concentration double optional !
in case, that default material is a ternary
pressure double optional !
lattice-constants
double_array optional !
lattice-constants-temp-coeff double_array optional !
crystal-type character optional ! required for
nondefault substrate material: e.g.
zincblende, wurtzite
wurtzite-rotation-matrix character optional !
$end_domain-coordinates
required !
!----------------------------------------------------!
Description
domain-type = l m n
l={0,1} ,
m={0,1} , n={0,1} ,
l+m+n=dimension (see
$simulation-dimension)
This specifier contains redundant information, i.e. the same information
as $simulation-dimension, orientation = ...
which
is now used.
However, the specifier domain-type
must be still there (CHECK: WHY???).although the values specified here can be arbitrary.
Domain extension - Minimum and maximum coordinates:
x-coordinates = xmin xmax
y-coordinates = ymin ymax
z-coordinates = zmin zmax
Only entries for the coordinate axes specified by orientation.
min, max
are double precision reals.
x-coordinates = 0.0d0 350.0d0
Crystal rotations with respect to simulation coordinate system
If hkl-x-direction-zb, ..., hkil-x-direction-wz,
... are not specified in the input file, the default directions
miller-default-direction-of-x, miller-default-direction-of-y
as specified in the database in $zb-restrictions
or $wz-restrictions
are used. This default coincides with the crystal fixed cartesian
coordinate system.
Zincblende
The orientation of the crystal with respect to the general three-dimensional
simulation coordinate system is fixed by the specification of the Miller
indices of two perpendicular planes.
The Miller indices define a plane. There exists a vector that is perpendicular to this plane, e.g. in zinc blende blende materials, the [hkl]
vector is always perpendicular to the (hkl) plane.
However, for wurtzite, this is not necessarily true. For instance, although the [0001] vector is perpendicular to the (0001) plane, in general is does not hold that the vector that is perpendicular to the (hkil) plane is
given by [hkil].
hkl-x-direction-zb = 0 1 0 !
Specify (hkl) plane perpendicular to x axis. In zinc blende the x
axis is then along this [hkl] direction. Here, (hkl) = (010).
hkl-y-direction-zb = 0 0 1 !
Specify (hkl) plane perpendicular to y axis. In zinc blende the y
axis is then along this [hkl] direction. Here, (hkl) = (001).
In this example, the x-axis of the simulation coordinate system is directed
perpendicular to the (0 1 0) plane,
i.e.
along [010] of the zinc blende crystal (crystal coordinate
system).
The y-axis of the simulation coordinate system is directed perpendicular
to the (0 0 1) plane, i.e.
along along [001]
of the
zinc blende crystal. The direction of the third axis
is calculated internally.
Default is:
hkl-x-direction-zb = 1 0 0 !
x axis perpendicular to (1 0 0) plane,
i.e. parallel to [100] direction
hkl-y-direction-zb = 0 1
0
!
y axis perpendicular to (0 1 0)
plane, i.e. parallel to [010] direction
(hkl-z-direction-zb = 0 0 1) !
0 0 1)
plane, i.e. parallel to [001] direction
hkl-z-direction-zb = 3 1
1 ! x axis perpendicular to (3 1
1) plane, i.e. parallel to [311] direction
hkl-y-direction-zb = 0 1 -1 ! 0 -1 -1)
plane, i.e. parallel to [0-1-1] direction
(hkl-x-direction-zb
In zinc blende crystals, the cation (metallic elements) polarity is labeled
{111}A, and the anion polarity is {111}B, which is {11-1}:
Wurtzite
Similar as above for wurtzite.
However, wurtzite is hexagonal and the cartesian axes system is cubic. For that
reason, some crystallographic equivalent planes will have different indices.
This can be solved using 4-digit Miller-Bravais indices (hkil) (i.e. four-axis a1,
a2, a3, c with a1, a2, a3
lying in the same plane (=basal plane)).
Rule: The first three indices must some to zero, i.e. h + k + i = 0
hkil-x-direction = 1 0 -1 0
! The x-axis of the simulation coordinate system is parallel to the vector that is perpendicular to
the (h k i l) = ( 1 0 -1 0
) plane.
hkil-y-direction = -1 2 -1 0
! The y-axis of the simulation coordinate system is parallel to
the vector that is perpendicular to the (h
k i l) = (-1 2 -1 0
) plane.
hkil-z-direction = 0 0
0 1 !
(h
k i l) =
( 0 0 0 1
) plane. (calculated internally)
The direction of the third axis is calculated internally.
Usually for wurtzite, the four-digit Miller-Bravais (or Bravais-Miller) indices (h k i l)
are used where i = - (h + k), i.e. i is not
independent.
The four integer values (Miller indices) that are given for hkil-x-direction refer to a
plane. The x direction is then the one that is perpendicular to this plane. This vector along the x axis has indices that are in general
not identical to the Miller indices hkl in wurtzite.
Why are four-digit Miller-Bravais indices used in wurtzite? This scheme for
labeling planes in a hexagonal lattice makes permutation symmetries apparent,
e.g. the similarity between (110) = (11-20) and (1-20) = (1-210) is more obvious
when the redundant index i is shown. A good reference is F. C. Frank, Acta Cryst. 18, 862 (1965).
Popular growth planes for wurtzite
- polar: c-plane: {0001}, Ga-terminated face (Ga-face or Ga
polarity)
- polar: {000-1}, N-terminated face (N-face
or N polarity)
- nonpolar: m-plane: {10-10},
theta = 90° (angle between growth direction and [0001] direction, i.e.
c-direction)
hkil-x-direction = 0 0 0 -1
! x coordinate axis is perpendicular to (0 0 0 -1)
plane
hkil-z-direction = 1 0 -1
0
! z coordinate axis is perpendicular to (1 0 -1
0) plane
If one grows the quantum well perpendicular to the (10-10) plane, then the pyroelectric and piezoelectric fields along the
growth direction z
are zero.
- nonpolar: a-plane:
{11-20}
- semi-polar: (1-101) plane
- semi-polar: (10-11) plane, Ga-face
- semi-polar: (10-13) plane, Ga-face
- semi-polar: (10-1-1) plane, N-face
- semi-polar: (10-1-3) plane, N-face
- semi-polar: (11-22) plane, Ga-face
hkil-x-direction = -1 -1 2 3
! x coordinate axis is perpendicular to (-1
-1 2 3) plane
hkil-y-direction = 1 -1 0
0 ! y coordinate axis is perpendicular to (1
-1 0 0) plane
! hkil-z-direction = ? ? ?
? ! is normal to (11-20) plane (This direction is not needed
in the input file. It is sufficient to specify only two axes: hkil-x-direction
and hkil-x-direction.)
The hkil-x-direction = -1 -1 2 3
forms an angle of about 32° with the vector perpendicular to the (0001)
plane.
a-plane and m-plane are orthogonal to each other.
Let's look at a further example. (This should be checked again.)
- semi-polar: (-1-12-2) plane
hkil-x-direction = 1 -1
0 0
! 1
-1 0 0) plane
hkil-y-direction = -1 -1
2 3
! y coordinate axis is perpendicular to (-1
-1 2 3) plane
hkil-z-direction = 8
8 -16 9 !
8 8 -16 9) plane
(1 -1
0 0)
plane is orthogonal to both (-1 -1
2 3) and (8
8 -16 9) planes
but the direction [-1 -1 2 3]
is not orthogonal to the direction [8 8 -16
9].
The reason for the latter is that the hexagonal unit cell in real crystals
does not fulfill the ideal c/a ratio, where it holds c/a = SQRT(3/8) =
1.63299....
a and c are the lattice constants.Assuming that the lattice constants
fulfill the ratio c/a, i.e.
%ideal_c = SQRT(8/3) * 0.3189
lattice-constants = 0.3189d0
0.3189d0 %ideal_c
! [nm] a,a,c
! lattice-constants = 0.3189d0
0.3189d0 0.52076151931570366367674259428247d0 !
[nm] a,a,c
then the orthogonality relation for hkil-z-direction =
8 8 -16 9 would
be fulfilled.
An alternative solution for the real c/a ratio would have
been:
hkil-z-direction = 100000000000
100000000000 -200000000000 113483513101 ! 113483513101 = 3 * a^2 / c^2.
- A user reported: "For example for (10-1-1) I use these 2
planes: (
1 -2 1 0) and (-178 0 178 101)."
When a simulation is performed, the screen output is redirected to a
.log file which has the same name as the input file.
This file contains some information about rotation of coordinate axes, e.g. for
the following settings
domain-type =
0 0 1
hkil-x-direction = 1 -1 0 0 !
x axis perpendicular to (1-100) plane
hkil-y-direction = 1 1 -2 -3 !
!hkil-z-direction = 1 1 -2 1 !
the following is written out:
....
wurtzite crystal orientation (Bravais-Miller indices as
specified in input file or database)
hkil-x-direction = 1.000000 -1.000000
0.000000 0.000000
hkil-y-direction = 1.000000
1.000000 -2.000000 -3.000000
wurtzite crystal orientation (Bravais-Miller indices [hkil]
are cooked down to 3 lattice axes used [hkl].)
hkl-x-direction = 1.000000 -1.000000
0.000000
hkl-y-direction = 3.000000
3.000000 -3.000000
3rd direction = 0.408248
0.408248 0.816497
3rd direction corresponds to (hkil) plane
= ( 0.408248 0.408248 -0.816497 0.816497)
$domain-coordinates: Getting lattice constants for substrate
material = GaN
substrate lattice constants for material 1 = 0.31890000
0.31890000 0.51850000 [nm]
c/a ratio in wurtzite = 1.625901537
(ideal ratio = 1.632993162).
3rd direction corresponds to [hkil] direction = [
0.408248 0.408248 -0.816497 0.463295]
angle between hkil directions =
90.000000000000 ° (substrate)
angle between hkil_1 and [0001] = 90.000000000000 °
(substrate)
angle between hkil_2 and [0001] = 148.406680335000 °
(substrate)
----------------------------------------------------------------------
Rotation matrix for material number = 1:
( 0.500000000 -0.866025404 0.000000000 ) (hkl-x-direction)
( 0.453699101 0.261943298 -0.851788022 ) (hkl-y-direction)
( 0.737670066 0.425894011 0.523886596 ) (hkl-z-direction)
(transformation matrix for a vector from crystal coordinate
system to simulation coordinate system: material_info(1)%transform_cxyz_to_xyz)
Eigenvalues:
( 0.142914947, 0.989734974) ==> e^(+i theta) =
cos(theta) + i sin(theta)
( 0.142914947, -0.989734974) ==> e^(-i theta) = cos(theta) -
i sin(theta)
( 1.000000000, 0.000000000) ==> eigenvalue = 1: axis of
rotation
Eigenvector: axis of rotation
( 0.645466750 0.000000000 )
( -0.372660402 0.000000000 )
( 0.666706007 0.000000000 )
Eigenvector: 2nd and 3rd
( -0.183290190 -0.508026617 )
( -0.656172319 0.000000000 )
( -0.189321403 0.491842410 )
( -0.183290190 0.508026617 )
( -0.656172319 -0.000000000 )
( -0.189321403 -0.491842410 )
sin(theta) = (0.9897349735464228,0.0000000000000000)
cos(theta) = (0.1429149472205782,0.0000000000000000)
theta = arcsin( sin(theta) ) = 81.783443016 ° = 0.454352461
[pi] = 1.427390354
theta = arccos( cos(theta) ) = 81.783443016 ° = 0.454352461
[pi] = 1.427390354
==> rotation angle = 81.783443016 ° = 0.454352461 [pi] =
1.427390354
==> rotation axis = [ 0.645466750 -0.372660402
0.666706007 ]
----------------------------------------------------------------------
material_orientation: l = 1
material_info(l)%interface_vector_in_cxyz = [ 0.737670066
0.425894011 0.523886596 ]
...
Homogeneous strain
The following specifier is only necessary for strain-calculation =
homogeneous-strain or hydrostatic-strain.
growth-coordinate-axis = 1 0 0 !
along x axis of simulation coordinate system
= 0 1 0 !
= 0 0 1 !
= l m n ! l={0,1} ,
m={0,1} , n={0,1} ,
l+m+n=1
growth-coordinate-axis = 0 0 1
hkl-z-direction-zb = 0 0 1
but it can be chosen to be oriented along any other crystal direction, e.g.
along the vector perpendicular to the (311) plane:
hkl-z-direction-zb = 3 1 1
The growth direction is only important for the calculation of pseudomorphic
strain as it enters the equation to calculate the strain tensor:
$simulation-flow-control
...
strain-calculation = hydrostatic-strain
! growth-coordinate-axis
strain-calculation = homogeneous-strain
! growth-coordinate-axis
! strain-calculation = homogeneous-strain-sim-system
! growth-coordinate-axis
If growth-coordinate-axis is not present, the default value
for 1D simulations is the simulation axis.
For 2D/3D simulations, the default value of
the
database_nn3.in file
is taken specified in
$domain-coordinates-defaults.
Specification on which material all layers are grown pseudomorphically.
This can
be either a material known from the database or a user defined name.
In the
latter case, an entry for the lattice constants and the crystal type of this
user defined material must be supplied.
In case of pseudomorphic growth on a
known ternary, its alloy concentration must be specified.
This specifier is not used for strain-calculation =
strain-minimization but it is used for strain-calculation =
strain-minimization-new.
In the former case, the lattice constant of the substrate is extracted from substrate-cluster-number
($strain-minimization-model).
pseudomorphic-on = GaAs
! If strain calculation is omitted, this specifier is optional. However,
for wurtzite, we still require it. (Should be fixed in the code.)
GaAs would be a known binary material, so no further input is required here.
Alloy concentration of ternary substrate (e.g. pseudomorphic-on
= Al(x)Ga(1-x)As)
alloy-concentration = 0.3d0 !
Al0.3Ga0.7As
alloy-concentration = x
! x={0.0d0,1.0d0}
If pseudomorphic growth is assumed to take place on an unknown material (e.g.
pseudomorphic-on = banana) the lattice constants of this material
and its crystal structure must be specified.
For the structure of the crystal,
only wurtzite and zinc blende are possible choices at the moment. In wurtzite,
the lattice constant orderering is a, a,
c.
lattice-constants = 0.4d0 0.4d0 0.4d0
lattice-constants-temp-coeff = 3.88d-6
3.88d-6 3.88d-6 ! [nm/K]
More information on temperature dependent lattice constants...
crystal-type =
zincblende ! also used for diamond-type materials
=
wurtzite
pressure = 9.4
! [GPa]
The substrate could be strained by an additional external pressure.
Rotation of wurtzite crystals
The ideal axial ratio of the hexagonal lattice (hexagonal close-packed structure) of c/a in wurtzite crystals is SQRT(8/3)=1.63299....
However, in real wurtzite crystals such as GaN, AlN and InN, the c/a ratio is
slightly different.
c is the lattice constant perpendicular the (0001) plane, a is the lattice
constant that lies in the (0001) plane.
In order to specify the rotation matrix that rotates the crystal
coordinate system to the simulation coordinate system, the Miller-Bravais
indices of the planes have to be specified (hkil-x-direction,
...), see the detailed description above.
wurtzite-rotation-matrix =
c-a-ratio-substrate
! Each material is using the rotation matrix of the common (or local)
substrate, i.e. substrate's c/a ratio. (default) ([Romanov2006]
seems to use the common substrate model.)
! (Note that nextnano³
allows to enter (optionally) for each material its own substrate, therefore the
term '(or local)' in the previous sentence.)
= c-a-ratio-ideal ! Each material is using the rotation matrix of the ideal wurtzite crystal. (This
is the implementation of nextnano++.) The rotation matrix for all materials is the same.
= c-a-ratio-local
!
For example, the angle between the plane (10-1-3) and the c plane (0001) is
different for (see e.g. [Romanov2006] A. E. Romanov et al., JAP
100, 023522 (2006))
- AlN (31.64°)
- GaN (32.04°)
- InN (31.81°) and
- the ideal wurtzite crystal (32.15°).
So if we specify hkil-x-direction = 1 0 -1 -3,
we have to define which c/a ratio we want to use.
c-a-ratio-substrate: By
default, for each material the c/a ratio of its global substrate is
used, i.e. the substrate's c/a ratio is used.
The crystal coordinate system has been rotated to the simulation coordinate system assuming the
substrate's c/a ratio.
Therefore, for all materials, the same rotation matrix is used (unless
one specifies for a material its own substrate).
c-a-ratio-ideal:
Optionally, the user can specify to use the ideal c/a ratio in order to be consistent with
the nextnano++ implementation.
The crystal coordinate system has been rotated to the simulation coordinate system assuming the ideal c/a ratio.
For all materials, the same rotation matrix is used.
More
information on how the rotation is carried out in the program ...
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