In general, the problem as defined in section Section 2.1.3 cannot be
solved analytically. Thus, the solution has to be calculated by numerical
methods, which normally require a discretization of the partial differential
equations. For that reason, the domain
where the equations are
posed has to be partitioned into a finite number of subdomains
,
which are usually obtained by a Voronoi tessellation
[156,33]. In order to obtain the solution with a desired accuracy,
the equation system is approximated in each of these subdomains by algebraic
equations. The unknowns of this system are approximations of the continuous
solutions at the discrete grid points in the domain [193].
Several approaches for the discretization of the partial differential equations
have been proposed. Due to the discretization of the current continuity
equations
it has been found to be
advantageous to apply the finite boxes discretization scheme for semiconductor
device simulation [193]. This method considers the equation integral
form for each subdomain, which is the so-called control volume
associated with the grid point
.
Before the Gauss integral theorem can be applied, the fluxes (2.58)-(2.60) are inserted in the balance equations (2.61)-(2.63) (analogously to the drift-diffusion model above):
![]() |
(2.79) |
![]() |
(2.80) |
![]() |
(2.81) |
![]() |
(2.82) |
![]() |
(2.83) |
![]() |
(2.84) |
![]() |
(2.85) |
![]() |
(2.86) |
![]() |
(2.87) |
Grid points on the boundary
are defined as having neighbor grid
points in other segments. Thus, (2.88) does not represent the complete
control function
, since all contributions of fluxes into the contact or
the other segment are missing. For that reason, the information for these boxes
has to be completed by taking the boundary conditions into account. Common
boundary conditions are the Dirichlet condition, which specifies
the solution on the boundary
, the Neumann condition,
which specifies the normal derivative, and the linear combination of these
conditions giving an intermediate type:
More complex models with exponential interdependence between the solution variables such as thermionic field emission interface conditions [185,201] have also been implemented.
A method has been under development to implement segment models calculating the interior fluxes and their derivatives independently from the boundary models. The segment models do not have to differentiate the point type, they do not even have to care about the boundary model used. The assembly system is responsible for combining all relevant contributions by using the information given by the boundary models.
To account for complex interface conditions, grid points located at the
boundary of the segments (see Figure 2.2a) have three values, one for
each segment (see Figure 2.2b) and a third point located directly at the
interface which can be used to formulate more complicated interface conditions
like for example, interface charges. However, to simplify notation these interface
values will be omitted in the discussion and only the two interface points,
and
, are used.
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Basically, the two (incomplete) equations
and
are
completed by adding the missing boundary fluxes
:
In the case of an arbitrary splitting of a homogenous region into different
segments, the boundary models have to ensure that the simulation results remain
unchanged. By adding (2.91) to (2.90), the box
of grid point
can be completed and the boundary flux is
eliminated. The merged box is now valid for both grid points, for that
reason the respective equation cannot only be used for grid point
, but also for
.
Whereas the segment models assemble the so-called segment matrix, the interface
models are responsible for assembling and configuring the interface system
consisting of a boundary and special-purpose transformation matrix. New
equations based on (2.92) can be introduced into the boundary
matrix without any limitations on , thus from 0
(Neumann) to
(Dirichlet). The interface
models are also responsible for configuring the transformation matrix to
combine the segment and boundary matrix correctly. Depending on the interface
type there are two possibilities:
As an additional feature, the transformation matrix can be used to calculate
several independent boundary quantities by combining the specific boundary
value with the segment entries (also in the case of Dirichlet
boundaries). For example, the dielectric flux over the interface is calculated
as
and introduced as a solution variable because some interface models
require the cross-interface electric field strength to model effects such as
tunnel processes. Calculation of the normal electric field is thus
trivial. Note that this is not the case when the normal component of the
electric field
has to be calculated using neighboring points when
unstructured two- or three-dimensional grids are used.
See Figure 2.3 for an illustration of these concepts. The transformations are set up to combine the various segment contributions with the boundary system.
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Contacts are handled in a similar way to interfaces. However, in the contact
segment there is only one variable available for each solution quantity
(
). Note that contacts are represented by spatial
multi-dimensional segments. Furthermore, all fluxes over the boundary are
handled as additional solution variables
(for example, contact charge
for the Poisson equation, contact electron
current
for the electron continuity equations, or
as the contact heat flow).
For explicit boundary conditions one gets
At Schottky contacts explicit boundary conditions apply. The semiconductor
contact potential
is fixed and given as the difference of the metal
quasi-Fermi level (which is specified by the contact voltage
) and the metal workfunction difference
potential
.
For Dirichlet boundary conditions one gets
For example, at Ohmic contacts simple Dirichlet
boundary conditions apply. The contact potential
, the carrier contact
concentrations
and
, and in the energy-transport simulation case, the
contact carrier temperatures
and
are fixed. The metal
quasi-Fermi level (which is specified by the contact voltage
) is equal to the semiconductor quasi-Fermi level. With the
constant built-in potential
(calculated after [65]), the
contact potential at the semiconductor boundary reads
![]() |
(2.98) |
For Neumann boundaries the flux over the boundary is zero hence the equation assembled by the segment model is already complete.