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# Regions

## Basic geometry objects

To built up a geometry, there are - dependent on the dimension of the simulation to be performed - various basic geometry elements available.
These geometry elements are specified within the keyword` \$regions ` and can be clustered to a bigger object later on.

## General specifiers

`region-number = `

an integer number `>= 1`
Numbering must be unique. All region numbers together must form a dense set 1,2,3,....,maxnumber

`region-priority = `

an integer number `>= 1`
In case of overlapping regions, the region with higher priority (= higher numerical value) overwrites the region with lower priority.

`!--------------------------------------------------!`
```\$regions                                optional   !```
``` region-number       integer            required   ! ```an integer number to refer to geometry element
``` region-priority     integer            required   ! ```a positive integer to set overwriting priority, a smaller number
```                                                   ! ```can be overwitten by any higher
``` base-geometry       character          required   ! ```type of geometry object:
```                                                   ! ```3D:``` cuboid```,` obelisk`,``` cone```,` truncated-cone`,` semiellipsoid`,``` hexagonal-obelisk```,``` triangular-prism                                                    ! ```2D:``` rectangle```,` triangle`,``` trapezoid```,``` polygon```,``` semiellipse                                                    ! ```1D:``` line```
``` x-coordinates       double_array       optional   ! ```for` cuboid`,``` rectangle```,` line`:``` xmin```,` xmax`
``` y-coordinates       double_array       optional   ! ```for``` cuboid```,``` rectangle```,` line`:``` ymin```,` ymax`
``` z-coordinates       double_array       optional   ! ```for``` cuboid```,``` rectangle```,` line`:``` zmin```,` zmax`
``` base-coordinates    double_array       optional   ! ```for``` obelisk```,` cone`,``` truncated-cone```,` trapezoid`:```                                                    ! xmin xmax ymin ymax zmin zmax ```of``` obelisk ```(`cone`)` ` base plane
```                                                   ! ```(one pair must be equal)
``` top-coordinates     double_array       optional   ! ```for``` obelisk```,` cone`,``` truncated-cone```,` trapezoid`,``` semiellipsoid```:
```                                                   ! xmin xmax ymin ymax zmin zmax ```of``` obelisk ```(`cone`)` ` base plane
```                                                   ! ```(one pair must be equal)```                                                    ! x y z ```for``` semiellipsoid```
``` corner-coordinates  double_array       optional   ! ```for``` triangle```: ``` x1 y1  x2``` `y2````  x3y3 ``` ```                                                   ! ```triangle corner coordinates, interpreted via orientation and similar for``` polygon  semi-ellipse-base   double_array       optional   ! ```for``` semiellipse```: base line ellipse:
```                                                   ! ```two pairs of``` xmin,xmax ymin,ymax zmin,zmax ```- one pair equal numbers
``` semi-ellipse-top    double_array       optional   ! ```for``` semiellipse```: top coordinate pair: e.g. for coordinate orientation  (101)` -> `(x,z)

```\$end_regions                            optional   !```
`!--------------------------------------------------!`

## 3-dimensional objects (only possible in 3D simulations):

`cuboid`,` obelisk`,` cone`,``` truncated-cone```,` semiellipsoid`,``` hexagonal-obelisk```,``` triangular-prism```

## 2-dimensional objects (only possible in 2D simulations):

`rectangle`,` triangle`,` trapezoid`,``` polygon```,` semiellipse`

`line`

# Details of specification

## 3-dimensional objects

### Cuboid

`\$regions`
` region-number     = 1`
` base-geometry     = cuboid`
` region-priority   = 1`
` x-coordinates     = xmin  xmax`
` y-coordinates     = ymin  ymax`
` z-coordinates     = zmin  zmax`
`\$end_regions`

The surfaces of the` cuboid `are assumed to be in coordinate planes of the simulation coordinate system. The coordinates above specifiy the six coordinate planes which limit the cuboid.

### Obelisk

`\$regions`
` region-number      = 1`
` base-geometry      = obelisk`
` region-priority    = 1`
` base-coordinates   = xmin xmax  ymin ymax  zmin zmax`
` top-coordinates    = xmin xmax  ymin ymax  zmin zmax`
`\$end_regions`

Base and top plane of the obelisk have to be in parallel coordinate planes. These planes are identified by the implicit rule, that a pair of coordinate values (e.g.  `ymin ymax`) has the same value (```ymin = ymax```). In this example, the plane is in the (x,z)-coordinate plane. The remaining four coordinates specify a rectangle in the corresponding plane.

### Truncated cone and cylinder

`\$regions`
` region-number      = 1`
``` base-geometry      = truncated-cone                    ! ```can be used to specify a cylinder
` region-priority    = 1`
` base-coordinates   = xmin xmax  ymin ymax  zmin zmax`
` top-coordinates    = xmin xmax  ymin ymax  zmin zmax`
`\$end_regions`

A cone with its apex cut off by a plane is called a truncated cone. In our implementation, the truncated cone is bounded by two ellipses of different size that are aligned parallel to each other.

Base and top plane of the truncated cone have to be in parallel coordinate planes. This plane is identified by the implicit rule, that a pair of coordinate values (e.g.  `xmin xmax`) has the same value (```xmin = xmax```). In this example, the plane is in the (y,z)-coordinate plane.  `ymin ymax`  and``` zmin zmax```  specify the diameter of the truncated cone top and base in the y and z direction, respectively. This corresponds to the specification of ellipses in the base and top plane.

How to specify a cylinder?

A cylinder is specified as a special case of a``` truncated-cone ```where the two boundary planes are circles. For a truncated cone, one specifies base and top coordinates. Let us assume we have a spherical cylinder of diameter 10 nm and height 15 nm. Then the base and top coordinates would be, for example,
` base-coordinates   = `
``` 10d0 20d0   10d0 20d0  15d0 15d0``` `  ! `(xmin,xmax,ymin,ymax,zmin,zmax) = (10,20,10,20,15,15)
``` top-coordinates    = 10d0 20d0   10d0 20d0  30d0 30d0``` `  ! `(xmin,xmax,ymin,ymax,zmin,zmax) = (10,20,10,20,30,30)
In other words, the x and y-coordinates specify the principal axes of the bottom and top ellipse (or circle) of the cylinder, respectively, and the z-coordinates specify the planes in which these two ellipses lie.

### Cone

`\$regions`
` region-number      = 1`
` base-geometry      = cone`
` region-priority    = 1`
` base-coordinates   = xmin xmax  ymin ymax  zmin zmax`
` top-coordinates    = x y z`
`\$end_regions`

Base plane of the cone has to be in parallel to the coordinate system planes. This plane is identified by the implicit rule, that a pair of coordinate values (e.g.  `xmin xmax`) has the same value (```xmin = xmax```). In this example, the plane is in the (y,z)-coordinate plane.  `ymin ymax`  and``` zmin zmax```  specify the diameter of the cone base in the y and z direction, respectively. This corresponds to the specification of an ellipse in the base plane. The top of the cone, the apex, is defined by the point (x,y,z).

The cone does not have to be right circular (where circular means that the base is a circle and right means that the axis passes through the center of the base at right angles to its plane).
Oblique cones are allowed, in which the axis does not pass perpendicularly through the center of the base.

The following figure shows several cones and truncated-cones (including the special case of cylinders). The related input file is called` 3D_cone.in`.

The cones are defined as follows:

``` !------------------------------------------------------------------------------- ! This is a cylinder. The base and top planes are parallel to the (y,z) plane. !------------------------------------------------------------------------------ region-number    = 1    base-geometry = truncated-cone     region-priority = 1      ! cylinder base-coordinates =  80d0 80d0     -80d0 -60d0     -80d0 -60d0                       ! xmin xmax ymin ymax zmin zmax top-coordinates  =  40d0 40d0     -80d0 -60d0     -80d0 -60d0                       ! xmin xmax ymin ymax zmin zmax  ```

``` !------------------------------------------------------------------------------- ! This is a right circular cone. The base plane is parallel to the (y,z) plane. !------------------------------------------------------------------------------- region-number    = 2    base-geometry = cone               region-priority = 1      ! cone (right circular) base-coordinates = -80d0 -80d0     50d0 90d0     50d0 90d0                          ! xmin xmax ymin ymax zmin zmax top-coordinates  = -10d0 70d0 70d0                                                  ! x y z !-------------------------------------------------------------------------------------------------------- ! This is a cone where the projection of the apex onto the base plane is located outside the base plane. ! The base plane is parallel to the (y,z) plane. !-------------------------------------------------------------------------------------------------------- region-number    = 3    base-geometry = cone               region-priority = 2      ! cone base-coordinates = -10d0 -10d0    -60d0 -20d0    -80d0 -20d0                        ! xmin xmax ymin ymax zmin zmax top-coordinates  =  40d0 -10d0 -10d0                                                ! x y z !-------------------------------------------------------------------------- ! This is a truncated cone. The base plane is parallel to the (y,z) plane. !-------------------------------------------------------------------------- region-number    = 4   base-geometry = truncated-cone     region-priority = 3      ! truncated-cone base-coordinates =  20d0 20d0      50d0 90d0      50d0 100d0                        ! xmin xmax ymin ymax zmin zmax top-coordinates  =  50d0 50d0      60d0 80d0      60d0  70d0                        ! xmin xmax ymin ymax zmin zmax !-------------------------------------------------------------------------------------------------------------------- ! These are three truncated cones where the projection of the top plane on the base plane is outside the base plane. ! The base plane is parallel to the (y,z) plane. !-------------------------------------------------------------------------------------------------------------------- region-number    = 5    base-geometry = truncated-cone     region-priority = 4      ! truncated-cone base-coordinates = -50d0 -50d0     20d0 40d0      20d0  60d0                        ! xmin xmax ymin ymax zmin zmax top-coordinates  = -20d0 -20d0     50d0 80d0      20d0  40d0                        ! xmin xmax ymin ymax zmin zmax```

### Semiellipsoid

`\$regions`
` region-number      = 1`
` base-geometry      = semiellipsoid`
` region-priority    = 1`
` base-coordinates   = xmin xmax  ymin ymax  zmin zmax`
` top-coordinates    = xtop ytop ztop`
`\$end_regions`

Base plane of the semiellipsoid must be in a coordinate plane. This plane is identified by the implicit rule, that a pair of coordinate values (e.g.  `ymin ymax`) has identical values (```ymin = ymax```).
In this example, the plane is in the (x,z)-coordinate plane. Top coordinates specify an arbitrary point "above" the ellipse, representing the base of the semiellipsoid.

Example: 3D sphere

A 3D sphere can be constructed from two semiellipsoids. In this example, the bottom planes of the two half-spheres are at z = 5 nm.
The upper half-sphere extends from 5 nm to 6 nm, the lower half-sphere from 5 nm to 4 nm.
The extensions in x and y directions for both half-spheres are from 4 nm to 6 nm.
Consequently, the sphere has a diameter of 2 nm.

``` region-number    = 1  base-geometry    = semiellipsoid                 ! ```(upper half-sphere)```  base-coordinates = 4d0 6d0   4d0 6d0   5d0 5d0   ! xmin xmax ymin ymax zmin=zmax  top-coordinates  = 5d0 5d0 6d0                   ! xtop ytop ztop  region-number    = 2  base-geometry    = semiellipsoid                 ! ```(lower half-sphere)```  base-coordinates = 4d0 6d0   4d0 6d0   5d0 5d0   ! xmin xmax ymin ymax zmin=zmax  top-coordinates  = 5d0 5d0 4d0                   ! xtop ytop ztop ```

### Hexagonal-obelisk

`\$regions`
` region-number      = 1`
``` base-geometry      = hexagonal-obelisk```
` region-priority    = 1`
` base-coordinates   = xmin xmax  ymin ymax  zmin zmax`
` top-coordinates    = xmin xmax  ymin ymax  zmin zmax`
`\$end_regions`

Base and top plane of the hexagonal-obelisk have to be in parallel coordinate planes. These planes are identified by the implicit rule, that a pair of coordinate values (e.g.  ```zmin zmax```) has the same value (```zmin = zmax```). In this example, the plane is in the (x,y)-coordinate plane. The remaining four coordinates specify a rectangle in the corresponding plane.
This geometry element is useful for wurtzite.
Many thanks to Lu Fu-Fa (Institute of Technology (CCIT), Taiwan, R.O.C.) for useful suggestions regarding the implementation of this geometry element.

Two` hexagonal-obelisk `shapes are possible:

• 'hexagonal-cylinder' with 6-fold rotational symmetry axis oriented along the z direction (i.e.``` zmin = zmax```).
Requirements: `xmin`(base)``` = ````xmin`(top)
```           xmax```(base)``` = xmax```(top)
```           ymin```(base)``` = ymin```(top)
```           ymax```(base)``` = ymax```(top)
Note: This condition should be fullfilled:``` ymax```(base)``` - ymin```(base)``` > (xmax```(base)``` - ``` `xmin`(base)`) / 0.866`
• 'hexagonal-pyramid' with 6-fold rotational symmetry axis oriented along the z direction (i.e.``` zmin = zmax```).
Requirements: `xmin`(top),` xmax`(top),``` ymin```(top),` ymax`(top) arbitrary
Note: This condition should be fullfilled:``` ymax - ymin > (xmax - ``` `xmin``) / 0.866`

For both shapes it holds:

• height of pyramid/cylinder:` zmax`(top)``` - zmax```(base)                  (`zmin =```` zmax```)
• width in x direction (distance between two parallel planes):``` xmax```(base)` - ``xmin`(base)
• width in y direction (distance between two corners) if the condition` ymax`(base)``` - ymin```(base)``` > (xmax```(base)``` - ``` `xmin`(base)`) / 0.866 `is fullfilled:
[`xmax`(base)``` - ````xmin`(base)]/0.866 [Note: 0.866 = cos(30°)]
If the above cited condition is not fullfilled, then the width is:``` ymax```(base)` - ``ymin`(base)
• width along y` > `width along x if` ymax`(base)``` - ymin```(base)``` > xmax```(base)``` - ``` `xmin`(base)
• center of hexagonal base plane on x axis: 0.5*(`xmin`(base)``` + xmax```(base))
• center of hexagonal base plane on y axis: 0.5*(`ymin`(base)``` + ymax```(base)):``` ymin```(base) and` ymax`(base) can be used to shift the hexagon along the y axis.

Two sides of the hexagonal base plane are aligned parallel to the y axis.
To rotate the hexagonal base plane by 30 degrees, the user has to specify values for `xmin`(top), `xmax`(top), `ymin`(top), ``` ymax```(top) so that it holds:
` xmax`(top)` - ` `xmin`(top)` > ymax`(top)``` - ymin```(top)

In this case it holds:

• width in x direction (distance between two corners) if the condition``` xmax```(base)``` - xmin```(base)``` > (ymax```(base)``` - ``` `ymin`(base)`) / 0.866 `is fullfilled:
[`ymax`(base)``` - ymin```(base)]/0.866 [Note: 0.866 = cos(30°)]
• width in y direction (distance between two parallel planes): ``` ymax```(base)``` - ymin```(base)
• width along y` < `width along x if``` xmax```(base)``` - xmin```(base)``` > ymax```(base)``` - y````min`(base)
• center of hexagonal base plane on x axis: 0.5*(`xmin`(base)``` + xmax```(base)): `xmin`(base) and `xmax`(base) can be used to shift the hexagon along the x axis.

If the 6-fold rotational axis is oriented along the x (i.e.``` xmin = xmax```) or y directions (i.e.``` ymin = ymax```), cyclic permutations hold for the above statements.

Example input file:` 3DHexagonalObelisk.in`

Screenshots:

Hexagonal shaped pyramid with flat top plane:

Hexagonal shaped pyramid:

Hexagonal shaped "cylinder":

 ``` ``` ``` ```

### Triangular prism

`\$regions`
` region-number      = 1`
``` base-geometry      = triangular-prism```
` region-priority    = 1`
``` corner-coordinates = x1 y1 z1   x2 y2 z2    x3 y3 z3                       x4 y4 z4   x5 y5 z5    x6 y6 z6```
`\$end_regions`

Example:
``` corner-coordinates = 10d0 10d0 10d0 ! x1 y1 z1                       10d0 30d0 10d0 ! x2 y2 z2                       20d0 20d0 10d0 ! x3 y3 z3                       10d0 10d0 40d0 ! x4 y4 z4                       10d0 30d0 40d0 ! x5 y5 z5                       20d0 20d0 40d0 ! x6 y6 z6```

Restrictions: ` triangular-prism ` must be oriented so that the triangles are perpendicular to either the x, y or z directions.
Example: Triangles perpendicular to z direction. Then it must hold:
```   ==> corner-coordinates => z1 = z2 = z3                              z4 = z5 = z6    ```In addition it holds:```                              x1 = x4, y1 = y4                              x2 = x5, y2 = y5                              x3 = x6, y3 = y6```

## 2-dimensional objects

### Rectangle

`\$regions`
` region-number     = 1`
` base-geometry     = rectangle`
` region-priority   = 1`
` x-coordinates     = xmin xmax`
` y-coordinates     = ymin ymax`
`\$end_regions`

Two pairs of delimiting coordinates are required. Whether these have to be x-coordinates and y-coordinates as in the example above, or another combination (e.g. x, z) depends on the simulation orientation which is specified already.

### Triangle (can also be specified using``` polygon```)

`\$regions`
` region-number      = 1`
` base-geometry      = triangle`
` region-priority    = 1`
` corner-coordinates = x1 y1    x2 y2    x3 y3`
`\$end_regions`

The corner coordinates refer to the plane, specified by the simulation orientation.

### Polygon (new)

`\$regions`
` region-number      = 1`
``` base-geometry      = polygon```
` region-priority    = 1`
``` corner-coordinates = x1 y1    x2 y2    x3 y3    x4 y4    x5 y5    ...```
`\$end_regions`

The corner coordinates of a polygon refer to the plane, specified by the simulation orientation.
The vertices may be listed clockwise or anticlockwise.
The first point could be repeated, i.e. the first point and the last point may be identical but this is not required.
The input polygon may be a compound polygon consisting of several separate subpolygons.
If so, the first vertex of each subpolygon must be repeated.

### Trapezoid (can also be specified using``` polygon```)

`\$regions`
` region-number      = 1`
``` base-geometry      = trapezoid```
` region-priority    = 1`
``` top-coordinates    = -15d0 15d0   -20d0 -20d0    ! xmin xmax  ymin ymax  base-coordinates   = -25d0 25d0   -40d0 -40d0    ! xmin xmax  ymin ymax \$end_regions```

For each,` base-coordinates `and` top-coordinates`, 2  values must be equal (meaning only 2 values can be equal), e.g.` ymin=ymax=-40d0`. Then one boundary of the object is a line at y=`-`40 nm. Base and top planes of the trapezoid must be in a coordinate plane.  This plane is identified by the implicit rule, that a pair of coordinate values (e.g.  `ymin ymax`) has identical values (```ymin = ymax```). This also implies that base and top lines are parallel to either the x or the y axis.
The example given above shows the blue trapezoid at the bottom of this figure.
The top line extends in x direction from -15 to 15 nm. The top line has a constant y coordinate of y = -20 nm
The base line extends in x direction from -25 to 25 nm. The base line has a constant y coordinate of y = -40 nm.

### Semiellipse

`\$regions`
` region-number      = 1`
` base-geometry      = semiellipse`
` region-priority    = 1`
``` semi-ellipse-base  = xmin xmax  ymin ymax  semi-ellipse-top   =    x1         y1 \$end_regions```

`semi-ellipse-base`: Here, 2 values must be equal (meaning only 2 values can be equal), e.g.` ymin=ymax=40d0`. Then one boundary of the object is a line at y=40 nm. Base plane of the semiellipse must be in a coordinate plane. This plane is identified by the implicit rule, that a pair of coordinate values (e.g.  `ymin ymax`) has identical values (```ymin = ymax```).

`semi-ellipse-top`: This defines a point which determines the height of the semiellipse. In our example (`ymin = ymax`), the plane is in the (x,z)-coordinate plane. Top coordinates specify an arbitrary point "above" the ellipse, representing the base of the semiellipse.
Top coordinate must be lower than upper coordinate of base line border in direction of axis mentioned above.
Top coordinate must be higher than lower coordinate of base line border in direction of axis mentioned above.
```xmin < x1 <xmax y1 < ymin=ymax ```or``` y1 > ymin=ymax```

Examples:

 ``` semi-ellipse-base  = xmin xmax  ymin ymax  semi-ellipse-top   =    x1         y1```  Example 1: ```semi-ellipse-base =  60d0 120d0  40d0 40d0 semi-ellipse-top  = 100d0        20d0```From these data, the following points are extracted:    (point:   `(x,y)`)```    ```base: ``` ( 60,40) (120,40)``` `   `top:```   (100,20)``` Example 2: ```semi-ellipse-base =  60d0 120d0  40d0 40d0 semi-ellipse-top  = 100d0        60d0``` We changed the y1 coordinate of` semi-ellipse-top `from``` 20d0 ```to` 60d0`. Example 3: ```semi-ellipse-base =  40d0 40d0  20d0 80d0 semi-ellipse-top  = 100d0       60d0``` Here we changed `semi-ellipse-base`: Now the two x coordinates have identical values. Example 4: ```semi-ellipse-base =  40d0 40d0  50d0 70d0 semi-ellipse-top  = 100d0       60d0``` Here we changed the ymin and ymax coordinates of``` semi-ellipse-base```: Now the y extension of the semiellipse is restricted from ymin = 50 nm to ymax = 70 nm. The baseline is at the fixed value for x = 40 nm. Example 5: ```semi-ellipse-base = 40d0 40d0 40d0 80d0 semi-ellipse-top  = 60d0 60d0``` ```semi-ellipse-base = 40d0 40d0 40d0 80d0 semi-ellipse-top  = 20d0 60d0``` Here we built a circle out of 2 semi-ellipses. Example 6: ```semi-ellipse-base = 42d0 42d0 40d0 80d0 semi-ellipse-top  = 60d0 60d0``` ```semi-ellipse-base = 38d0 38d0 40d0 80d0 semi-ellipse-top  = 20d0 60d0``` Same as example 5 but this time, we moved the baselines a little bit apart from each other to make example 5 easier to understand.

## 1-dimensional objects

### Line

`\$regions`
` region-number      = 1`
` base-geometry      = line`
` region-priority    = 1`
` x-coordinates      = xmin  xmax`
`\$end_regions`

Chosen coordinates must be consistent with simulation orientation.