Conservation of mass is the one most important property a discretization of the Wigner equation has to fulfill. This subsection discusses the constraints this imposes on the mesh.
The main difficulty in achieving a discrete conservation law lies in the discretization of the drift term
In the continuous infinite case the -integral over
vanishes
because
is odd in
and the integral appearing on the right hand
side of Equation 8.7
To discuss the discrete case we assume that the -domain used
for
has limits
and
. We denote the length of the integration
domain by
.
Then the integration argument extends from
to
and we integrate over
with limits
to
in the right hand side of
Equation 8.10.
The integral 8.11 vanishes in the case that
is not only odd but also periodic with
period
.
Then the integral 8.11 always extends over exactly
one period and hence vanishes in each case as it is independent
of
.
To achieve a periodic
in the case of discrete
one can introduce a special
mesh. This method was introduced by
Frensley [Fre90]. It assumes an arbitrary (possibly
shifted) equispaced
mesh for
in 8.9. Here we write the
-mesh in the unshifted form
![]() |
(8.12) |
![]() |
(8.13) |
![]() |
(8.14) |
This condition is known from [Fre90]
as a completeness condition. It links
the spacing of the -mesh in the calculation of the
Wigner potential with an apparent periode length
of
.
We now assume that
is odd in
and periodic with period
.
Then
in order that the discrete version of
![]() |
![]() |
(8.16) |
The meshing condition 8.15 fixes only the
spacing
to be used in the discrete
form of 8.9
but it does not fix the limits of the integral, i.e.,
the number of points
.
However, given an equispaced
-mesh for
with
points, it seems natural to choose
the mesh in such a way that definition 8.9
corresponds to
the expansion of an odd function into a series of sines.
Then the denomination ``completeness'' is also justified.
The discrete sine transform is
![]() |
(8.17) |
![]() |
(8.18) |
![]() |
(8.19) |
![]() |
(8.20) |
![]() |
We want to stress, that the -mesh used in the discretization
of the Wigner potential can in principle be chosen
independently and differently from the
-mesh, especially,
if the band is given analytically - for example,
in a flat band model.
In a self-consistent simulation
one normally uses the same mesh for
and for
.
In any case the
number
of points in
and the number of points
in
can be chosen independently and usually one chooses
slightly smaller than
.
In practice one starts with choosing an equispaced
mesh using points for the simulation domain.
Then one chooses a coherence length
which is a multiple
of the
-mesh spacing
,
see Equation 8.21.
The period length
of the
-mesh is then fixed
by the completeness condition Equation 8.15
as
![]() |
(8.22) |
![]() |
(8.23) |
The artificial introduction of a Wigner potential
which is periodic in
is an elegant trick
to obtain a discretization conserving mass. We also
experimented with meshes which do not obey this
completeness condition.
But the results from such
simulations were unusable for TCAD purposes. Hence meshing
is essentially restricted to the use of an equispaced
-product grid.
These discussions are in accordance with transformation-theoretic theorems stating that in a certain sense there are no reasonable discretizations of the Wigner transform. Every discretization lacks one or more properties characteristic of the continuous transform.
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