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doping-function

Doping profiles

Doping profiles can be specified by the product of n functions. n is the dimension of the simulation, i.e. n = 1, 2 or 3. Each function depends only on one coordinate.

The doping profile is independent of the regions specified before. The function is applied to the region given by the specifier only-region.

The doping concentration is at the position specified by the specifier position.
The function is normalized such that the result is doping concentration at position position.
The impurity number specifies the kind of impurities used in the profile.

Note: Can be used with a doping concentration sweep where the doping-concentration is varied stepwise.

 

!------------------------------------------------------------------!
$doping-function                                        optional   !
 doping-function-number                 integer         required   !
 impurity-number                        integer         required   !
 base-function-1                        character       optional   !
 base-function-2                        character       optional   !
 base-function-3                        character       optional   !
 apply-function-1-along-dir             integer_array   optional   !
 apply-function-2-along-dir             integer_array   optional   !
 apply-function-3-along-dir             integer_array   optional   !
 doping-concentration                   double          required   !
 position                               double_array    optional   !
 parameters-base-function-1             double_array    optional   !
 parameters-base-function-2             double_array    optional   !
 parameters-base-function-3             double_array    optional   !
 exclude-materials                      integer_array   optional   !
 only-region                            double_array    optional   !
                                                                   !
 doping-profile-defined-by-function     character       optional   !
new
                                                                   !
 read-in-doping-file                    character       optional   !
 doping-filename                        character       optional   !

                                                                   !
 doping-sweep-active                    character       optional   !

 doping-sweep-step-size                 double          optional   !
 doping-sweep-number-of-steps           integer         optional   !
$end_doping-function                                    optional   !
!------------------------------------------------------------------!

 

Example

Constant doping profile

 $doping-function

  doping-function-number = 1
  impurity-number        = 1               !
properties of this impurity type have to be here: $impurity-parameters
  doping-concentration   = 0.5d0           ! 0.5 * 10^18 cm^-3
  only-region            = 10.0d0 160.0d0

 $end_doping-function

Note: Can be used with a doping concentration sweep where the doping-concentration is varied stepwise.

 

 

doping-function-number = 1
                      
= 2
                      
= ...

An integer number. At the very end, the doping function numbers must be given in a way, that a dense ascending series starting at 1 can be formed.

impurity-number = 1
              
 = 2
              
 = ...

An integer number. Properties of this impurity number have to be specified later.
This is a reference to an impurity and its parameters which will be specified by $impurity-parameters.

 

More complicated doping profiles

You can use different base functions along each simulation direction to define more complicated doping profiles.

base-function-1 = string-1
base-function-2 = string-2
base-function-3 = string-3
a valid base function name
string-i can be one of the one-dimensional functions: constant, linear, gauss-1d, step-1d, well-1d
The final doping profile will result from a product of these functions.

parameters-base-function-1 = double1 ....
parameters-base-function-2 = double1 ....
parameters-base-function-3 = double1 ....
function parameters
Parameters for the selected base functions. Dependent on the base function chosen, the following is expected:

 

Constant base function: Constant doping profile

base-function-1            = constant
No further function parameters required in 1D simulations.

Note: Can be used with a doping concentration sweep where the doping-concentration is varied stepwise.

 

 

Linear base function: Linear doping profile

base-function-1            = linear
The example shows which additional parameters are necessary.

Example:

$doping-function

 doping-function-number     = 1                       !
doping function #1
 impurity-number            = 1                       !
properties of this impurity type have to be specified below
 doping-concentration       = 0.5d0                   ! 0.5 * 1.0 * 10^18 cm^-3 = 0.5 * 10^18 cm^-3
 position                   = -20d0                   !
doping concentration refers to that position, i.e. -20 nm
 only-region                = -20d0 0d0               !
only from -20 nm to 0 nm
 base-function-1            = linear                  !
linear doping profile
 apply-function-1-along-dir = 0 0 1                   !
along z direction
 parameters-base-function-1 = -20d0 0d0   0.5d0 0.0d0 ! (1) zmin = -20 nm      (2) zmax = 0 nm
                                                      ! (3) 0.5 * 10^18 cm^-3  (4) 0.0 * 10^18 cm^-3

 doping-function-number     =                     !
doping function #2
 impurity-number            = 2                       !
properties of this impurity type have to be specified below
 doping-concentration       = 1d0                     ! 1.0 * 10^18 cm^-3
 position                   = 10d0                    !
doping concentration refers to that position, i.e. 10 nm
 only-region                = 0d0 10d0                !
only from 0 nm to 10 nm
 base-function-1            = linear                  !
linear doping profile
 apply-function-1-along-dir = 0 0 1                   !
along z direction
 parameters-base-function-1 = 0d0 10d0    0.0d0 1.0d0 ! (1) zmin = 0 nm         (2) zmax = 10 nm
                                                      ! (3) 0.0 * 10^18 cm^-3   (4) 1.0 * 10^18 cm^-3

$end_doping-function

 

 

LSS theory (Lindhard, Scharff, Schiott theory) - Gaussian distribution of ion implantation impurity profile

base-function-1            = gauss-1d         ! LSS theory
parameters-base-function-1 = center-coordinate  gauss-width         minimum-value  maximum-value
center-coordinate
is the position of the Gauss center along the relevant direction i in units of [nm].
gauss-width
is usually called sigma in the formula of the Gaussian distribution function (in units of [nm]).
For the meaning of gauss-width have a look at the 10 DM banknote of the German "Deutsche Mark" or any mathematical textbook.
minimum-value minimum  value of doping concentration in units of 1018 [cm-3]
maximum-value maximum value of doping concentration in units of 1018 [cm-3]

Within LSS theory the specifiers correspond to the following notations:
base-function-1            = gauss-1d         ! LSS theory
parameters-base-function-1 = projected-range    projected-straggle  minimum-value  maximum-value
apply-function-1-along-dir = 0 0 1            !
along z direction
doping-concentration       = implanted-dose / ( SQRT(2*pi) * projected-straggle ) ! concentration at reference position (see below)
position                   = projected-range  !
doping concentration refers to this position

projected-range   
Rp (ion's projected range) in units of [nm], i.e. the depth where most ions stop.
projected-straggle Delta Rp (ion straggle) in units of [nm], i.e. the statistical fluctuation of Rp.
implanted-dose     phi in units of [1/cm^2] (dose of the implant), typical ranges are from 1d11 to 1d16.

The program calculates from these parameters the dopant distribution using LSS theory. The calculated dopant profile can be printed out using the keyword $output-material and the specifier doping-concentration.

The 1D Schrödinger-Poisson tutorial shows a figure of two impurity profiles based on LSS theory.



For further details see for example:
"Very brief Introduction to Ion Implantation for Semiconductor Manufacturing" by Gerhard Spitzlsperger

 

 

User-defined doping function

Using the specifier doping-profile-defined-by-function, one can define an arbitrary function n(x,y,z) for the doping profile.
The value of this function is finally multiplied by the value specified in doping-concentration. Obviously, one can also set doping-concentration = 1d0.
The variables x, y, z refer to the grid point coordinates of the simulation area in units of [nm].

Example: The red Gaussian shaped curve in the above figure can also be achieved by defining the Gaussian function directly:

$doping-function
 ...
   %mu      = 86                                      !    mu
of Gaussian distribution (DisplayUnit:nm)
   %sigma   = 44                                      ! sigma
of Gaussian distribution (DisplayUnit:nm)
   %max_dop = 0.181337400182469                       !
maximum doping concentration at center of Gaussian distribution (DisplayUnit:1e18cm^-3)

 doping-function-number             = 4               !
doping function #4
 impurity-number                    = 2               !
 doping-concentration               = %max_dop        ! 1 * 10^18 cm^-3
 doping-profile-defined-by-function = " exp(- (z-%mu)^2 / (2*%sigma^2) ) "  ! n(x,y,z) = ...  
(dimensionless)
                                    = no              !
do not use user-defined doping function

The following operators and functions are supported:
+ , - , * , / , ^
abs
, exp , sqrt , log , log10 , sin , cos , tan , sinh , cosh , tanh , asin , acos , atan

 

 

Predefined doping functions

base-function-1            = step-1d
parameters-base-function-1 = para(1) = center, para(2) = width, para(3) = leftval, para(4) = rightval

base-function-1            = well-1d
parameters-base-function-1 = para(1) = center, para(2) = width, para(3) = leftval, para(4) = rightval
                             para(5) = center, para(6) = width, para(7) = leftval, para(8) = rightval
First 4 parameters for left step, second 4 parameters for right step.
This is a well with double gauss walls. The walls are centered at parameter center and the slope of the walls is given by width.

apply-function-1-along-dir = i j k
apply-function-2-along-dir = i j k
apply-function-3-along-dir = i j k
Variation of function-i is along the specified direction (0 0 1 or 0 1 0 or 1 0 0).

doping-concentration = double
concentration at reference position (see below).
A doping concentration at the position specified by the next specifier (position). The function defined above is normalized such that the result is doping concentration at position position.

Example:
We take
a constant doping with a concentration 8.0*1018 cm-3.

1D simulation: 8.0d0 * 1018/cm3.
   doping-concentration =
8.0d0

2D simulation: 8.0d0 * 1018/cm3.
  
doping-concentration = 8.0d0

3D simulation: 8.0d0 * 1018/cm3
   doping-concentration = 8.0d0

So we take the value to be 8.0d0 because we assume a 3D doping although we do a 1D or 2D simulation.
Thus it would be wrong to take
- 2.0
*106 cm-1     in the 1D case (cubic root)
- 4.0
*1012 cm-2   in the 2D case (squared cubic root)

position = coord1 ...
Doping concentration refers to that position.
The coordinates of a position which is used to fix normalization of the doping function profile. Can be omitted only for constant doping.

exclude-materials = num1 ...
To keep certain materials free from doping (e.g. air).
A list of defined material numbers which should not be doped.

only-region = coord1 ...
Apply doping function only to this region (coordinates of a cube, rectangle, line).
Restrict doping to this region only. The region is either a cube, rectangle or a line. The coordinates given specify the extension of the region as usual.
Note: See comments on how to specify correct interfaces further below.

 

Example (2D):

$doping-function                                     !
                                                     !
 doping-function-number = 1                          !
 impurity-number        = 1                          !
properties of this impurity type have to be specified by $impurity-parameters
 doping-concentration   = 10d0                       !
10 * 1018/cm3.
 only-region            = 0.0d0 50.0d0 30.0d0 60.0d0 ! xmin xmax  ymin ymax
                                                     !
$end_doping-function                                 !

 

Note: It you want to generate a very "accurate" doping profile, then you should apply the doping between interfaces. Interfaces are set if the material number is different.
Example: You want to specify a doping between 35.0 and 35.3 nm. Then you should consider to define a separate region, a separate cluster and a separate material for this 0.3 nm region.

Accurate doping profile:
   35.0     35.15    35.3                 ! [nm]
     x        x        x        x         !
physical grid points
     |                 |                  !
interface
    o|o       o       o|o       o         !
'multiple grid point' grid points (including multiple points)
.         .       .         .             ! material grid points (material parameters)
----- _________________ ------------------!
doping area
 
   0|1       1       1|0                 !
weighting factor
 

Not so accurate doping profile:
   35.0     35.15    35.3                 ! [nm]
     x        x        x        x         !
physical grid points
     |                                    !
no interface at 35.3 nm
    o|o       o        o        o         !
'multiple grid point' grid points (including multiple points)
.         .       .         .             ! material grid points (material parameters)
----- ____________________ ---------------!
doping area
    0|1       1       0.5                 !
weighting factor (an average)
 

Another not so accurate doping profile:
   35.0     35.15    35.3                 ! [nm]
     x        x        x        x         !
physical grid points
                                          !
no interfaces at all
     o        o        o        o         !
'multiple grid point' grid points (including multiple points)
.         .       .         .             ! material grid points (material parameters)
-- _______________________ ---------------!
doping area
   
0.5       1       0.5                 !
weighting factor
 

Obviously all three cases can produce different results.

 

Example (3D):

The following figure shows a 3D doping profile that is defined inside a 20 nm x 20 nm x 50 nm cube where the 50 nm are the z direction. The doping profile is homogeneous with respect to the (x,y) plane, it only varies along the z direction.

The doping profile is constant between z = 10 nm and z = 25 nm with a concentration of 1 x 1018 cm-3. It has Gaussian shape from z = 25 nm to z = 45 nm. It is zero between z = 0 nm and z = 10 nm, as well as between z = 45 nm and z = 50 nm.

 $doping-function

  !-------------------------------------
  ! constant doping of 1 * 10^18 cm^-3.
  !-------------------------------------
  doping-function-number     = 1                                 !
first doping funtion
  impurity-number            = 1                                 !
  doping-concentration       = 1.0d0                             ! 1.0d0 => 1.0 * 10^18 / cm^3
  only-region                = 0d0 20d0  0d0 20d0  10.0d0 25.0d0 ! xmin xmax  ymin ymax  zmin zmax


  !--------------------------------------------------------------------------
  ! Gaussian shaped doping along z direction, constant doping in (x,y) plane
  !--------------------------------------------------------------------------

  doping-function-number     = 2                                 !
second doping function
  impurity-number            = 1                                 !
  only-region                = 0d0 20d0  0d0 20d0  25.0d0 45.0d0 ! xmin xmax  ymin ymax  zmin zmax

  base-function-1            = constant                          !
  base-function-2            = constant                          !
  base-function-3            = gauss-1d                          !
  apply-function-1-along-dir = 1 0 0                             !
constant doping along x direction
  apply-function-2-along-dir = 0 1 0                             !
constant doping along y direction
  apply-function-3-along-dir = 0 0 1                             !
Gaussian shaped doping along z direction

  doping-concentration       = 1d0                               ! 1.0d0 => 1.0 * 10^18 / cm^3
  position                   = 10d0 10d0 25d0                    !
doping concentration refers to that position
  parameters-base-function-1 = 0d0 100d0
  parameters-base-function-2 = 0d0 100d0
  parameters-base-function-3 = 25d0 6d0  0d0 1.0d0               ! center-coordinate  gauss-width  minimum-value  maximum-value

 $end_doping-function

If you want to obtain the input file that was used to obtain this 3D doping profile plot (constant + Gaussian shape), please contact stefan.birner@nextnano.de.
-> 3Ddoping_profile.in

 


Reading in doping profiles from a file

read-in-doping-file = yes/no
-
flag for reading in doping profile from a file
- valid for n-type and p-type doping
- valid for arbitrary doping-function-number
-
can be combined with explicitly specifying doping profile by input file and by reading in doping profile from several files
- all doping functions will be added to the previously specified values (like superpositions)

Restrictions:
- The value of this profile is finally multiplied by doping-concentration. Obviously, one can also set doping-concentration = 1d0.
- You can specify a region (only-region) for which the doping file applies to, the outer region will not contribute.

doping-filename = doping_input.dat
                = doping/doping_input.dat
doping filename to be read in (e.g. experimental values). The string can include a folder name.
The ASCII file must contain 2 (1D), 3 (2D) or 4 (3D) columns in each line:

1D:
x coordinate [nm]                                            doping concentration [1*1018 cm-3]
...                                                          ...

2D:
x coordinate [nm]    y coordinate [nm]                       doping concentration [1*1018 cm-3]
...                  ...                                     ...

3D:
x coordinate [nm]    y coordinate [nm]   z coordinate [nm]   doping concentration [1*1018 cm-3]
...                  ...                 ...                 ...

The first line of this ASCII file can contain an optional header line with column descriptors.

If you want to obtain an input file that shows how to import a doping profile from a file
(1D_read_in_potential_and_doping_profiles.in,
 2D_read_in_potential_and_doping_profiles.in,
 3D_read_in_potential_and_doping_profiles.in),
please contact stefan.birner@nextnano.de.

 


Doping concentration sweep

It is possible to sweep over the doping concentration, i.e. to vary the doping concentration stepwise.
In each doping sweep step, the specifier doping-concentration is increased by doping-sweep-step-size in units of [1*1018 cm-3].
The output is labelled by ..._ind000.dat, ..._ind001.dat, ... for each doping sweep step.

 doping-sweep-active           = yes       ! to switch on doping sweep
                               = no        !
to switch off doping sweep
 doping-sweep-step-size        = 0.2d0     ! increase doping concentration in each step by ... in units of [1*1018 cm-3]
                                           !
(This value can also be negative.)
 doping-sweep-number-of-steps  = 5         !
number of doping sweep steps

Restrictions:
- Voltage sweeps ($voltage-sweep) and other sweeps cannot be combined with doping sweeps at present.
- Only one doping sweep is allowed at present.