Specifying the Dopant Distribution
The distribution of dopants within the semiconductor can be specified using the Analytic Doping Model and Geometric Doping Model features. The Analytic Doping Model enables distributions to be defined in terms of the coordinate system; the Geometric Doping Model enables definitions in terms of the distance from selected boundaries in the geometry.
Theory for the Analytic Doping Model
The Analytic Doping Model has two options to specify the doping concentration in terms of the coordinate system — User defined or Box. It supports the use of a local coordinate system that is rotated relative to the global coordinates.
User Defined Profiles
For user defined profiles any arbitrary dopant concentration can be created with user-defined functions or imported data (which can be specified using an interpolation function). Any expression written in terms of the local coordinate system can be used to define the distribution. Use of the COMSOL Multiphysics built-in functions enables the creation of intricate distributions.
Box Profiles
The Box profile enables a box-shaped region of uniform doping to be defined away from which the dopant concentration decays with one of three preset functions. The location of the region of constant doping is defined by specifying either the corner or center coordinate using the global coordinate system. The size of the region is defined by specifying its extent in each of the directions parallel to the local coordinate axis. In 3D the height, width, and depth are supplied to create a block; in 2D the width and depth are supplied to create a rectangle; and in 1D the width is supplied to create a line. It is possible to set the constant region to have zero extend in a given direction to, for example, create a plane of constant doping in a 3D geometry. The orientation of the constant region is aligned with the local coordinate axes. If a rotated local coordinate system is used the constant region rotates around its specified corner or center coordinate, which is expressed using the global coordinate system.
The distribution outside of the constant region decays with one of three preset functions: Gaussian, Linear, or Error Function. These functions are defined using Ramp functions with unity gradient that begin at the edges of the constant region and which continue throughout the remainder of the domain. Thus there are two Ramp functions for each geometry axis. Figure 3-6 shows the four Ramp functions used to create a 2D Box profile. Note how each Ramp function is zero inside the constant region, and that the unity gradient outside the region creates and effective coordinate axis for each direction that is zeroed at the region boundary.
In 3D, the Gaussian distribution is given by
(3-43)
where Nd,a is the donor or acceptor concentration, N0 is the concentration inside the uniformly doped region, li is the Decay length in the i-direction, and rx-, rx+, ry-, ry+, rz-, and rz+ are the Ramp functions.
Functions and Ramp in the COMSOL Multiphysics Reference Manual.
The decay length can be specified directly or calculated from a specified junction depth via:
where dj,i is the junction depth in the i-direction, and Nb is the background doping concentration that can be entered directly or taken from the output of another doping feature in the model. By default dj,x=dj,y=dj,z=dj, such that lx=ly=lz. To use different decay lengths in different directions select the Specify different length scales for each direction check box under the Profile section.
The linear distribution is given by
(3-44)
where gi is the gradient in i-direction. The gradient can be supplied directly or calculated from a specified junction depth via:
By default the gradient is the same in all directions, however it can be set to be direction dependent using the Specify different length scales for each direction check box. Note that a negative dopant distribution is not physical, so the concentration is set to zero in regions where Equation 3-44 gives Na,d<0.
The Error Function distribution is given by
where mx is an argument factor which controls the length scale of the profile. The argument factor can be entered directly or calculated from a specified junction depth via:
By default the argument factor is the same in all directions, however it can be set to be direction dependent using the Specify different length scales for each direction check box.
Figure 3-6: Ramp functions used to specify box doping profiles on a rectangular domain in 2D. The height out of the plane shows the magnitude of the ramp function (not to scale). Top: Individual ramp functions for the x-direction. Middle: Individual ramp functions for the y-direction. Bottom: Composite of all the ramp functions, showing the region of constant doping (highlighted in red).
Theory for the Geometric Doping Model
The Geometric Doping Model feature enables doping distributions to be defined in terms of the distance from selected boundaries in the geometry. This is convenient when working with geometries with intricate shapes that would be challenging to describe analytically using the coordinate system. The boundaries from which the distance is calculated are selected using the Boundary Selection for Doping Profile node. Any boundary that bounds, or is within, the domains to which the corresponding Geometric Doping Model feature is applied can be selected. The form of the distribution can be selected from a range of preset functions or a user defined expression can be supplied.
User Defined Profiles
When a user defined profile is selected any arbitrary distribution that can be written in terms of the distance from the selected boundaries can be defined. This distance is available as the variable semi.gdm#.D, where # corresponds to the number of the Geometric Doping Model feature.
Preset Profiles
The other selections in the Dopant profile away from the boundary list allow either Gaussian, Linear, or Error Function profiles to be generated. These profiles are defined in terms of the distance, D, from the selected boundaries as described below.
The Gaussian profile is given by
where Na,d are the concentration of the acceptors or donors, N0 is the concentration of dopants at the selected boundaries, and l is the decay length of the Gaussian function. The decay length can be entered directly or can be calculated from a specified junction depth, dj, via:
where Nb is a specified background doping concentration that can either be directly defined or taken from the output of another doping model feature.
The Linear profile is given by
(3-45)
where g is the gradient, which can be entered directly or calculated from a specified junction depth via
Note that, as a negative concentration is not physical, the dopant concentration is set to zero in regions where Equation 3-45 gives Na,d<0.
The Error Function profile is given by
where m is an argument factor which controls the length scale of the profile. The argument factor can be defined directly or calculated from a specified junction depth via:
Using Analytic and Geometric Doping Features Together
A doping distribution can be created by combining the functionality of the two doping model features. Often it is desirable to specify a domain of constant doping using a user defined profile in the Analytic Doping Model feature, and then to set the profile away from this region using the Geometric Doping Model feature. This approach has the advantage that the constant user defined profile can be assigned to a domain of any shape, thus removing the requirement for the uniformly doped region to be block-shaped that is imposed by a Box profile. The use of the Geometric Doping Model to specify the profile away from the constant domain has the advantage that it can accommodate a constant region that has curved boundaries. This is because the Geometric Doping Model profile only depends on the distance from the selected boundaries, rather than on the coordinate system.