In our application the environment
(electrodes) absorbs outgoing particles and injects particles
into the device
with a Fermi-Dirac equilibrium distribution which is fixed
by the demand of local charge neutrality at the boundary.
Adding boundary conditions
can change the character of the model and
in general a separation ansatz
will no longer work.
However, if we have an open system and
incoming waves are mixed states
built from plane waves of the form
then this type of boundary conditions can still be treated by
the separation
approach.
In this way the von Neumann equation with absorbing boundary conditions
is equivalent to a countable system of
Schrödinger equations with transparent boundary conditions.
In practice we solve the open Schrödinger equation for a set of
values for .
Then the solution is built from the pure states
which
are summed with the correct weight given by
the equilibrium Fermi-Dirac distribution in the electrodes.
Solving the von Neumann equation by the separation method
is known
as the quantum transmitting boundary method. It has been
introduced in [LK90b].
It has been applied for device simulations in [BA00],
and [Fre87]. For an extension see [LKF02].
The reflection and transmission coefficients for pairs of energy-related incoming waves from left and right are not independent [Akt99] which can be used to reduce the numerical costs by a factor of 2.
Ballistic transport is only a good approximation if the influence of scattering can be neglected. The catch here is that in such cases the assumption of injection with an equilibrium distribution is doubtful.
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