The idea of transparent boundary conditions is to model the problem in the unbounded domain. Then one imposes boundary conditions such that the solution from the finite problem is exactly the same as the solution from the unbounded problem restricted to the simulation domain.
We now apply this procedure to the stationary Schrödinger equation
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The device region is represented by the spatial interval .
We assume that
is constant in the leads:
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|
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Part of the incoming plane wave is reflected by the potential
and goes back to :
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(7.2) |
Note that
(and not
) has to fulfill
the absorbing boundary conditions, i.e., we really have inhomogeneous transparent
boundary conditions on the left boundary.
The other part of the wave is transmitted and travels to .
With
in the left lead the energy of the incoming particle
is
. Using the fact that energy
is conserved we get
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(7.3) |
In the left lead the steady state solution hence takes the form
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(7.4) |
The three subproblems (one for each domain) are coupled by the assumption
that ,
are continuous across the artificial boundaries at
and
.
This allows us to eliminate the a-priorily unknown coefficients
and
.
For a fixed wave vector
of the incoming wave this yields the
boundary value problem (BVP):
We note that the boundary conditions contain the spectral
parameter and are non-homogenous as the amplitude
of the incoming wave packet is normalized to unity.
The Fourier transform of the wave
exists only in
the sense of distributions.
It is important to find a good numerical discretization of the transparent BCs. This is done in [Arn01] where the discretization is determined in such a way that there are no reflections in the case of a flat potential and a homogenous incoming wave.
We have only derived transparent boundary conditions for the
stationary case. The same can also be done for the transient
Schrödinger equation.
However in this case the boundary conditions
become non-local in time and the equation
``has a memory''. Recently Ehrhardt
and Arnold [AES03] derived
a discretization which they claim to be feasible
for numerical simulation.
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