Solving the Schrödinger-Poisson System
The Schrödinger-Poisson system is special in that a stationary study is
necessary for the electostatics, and an eigenvalue study is necessary for the
Schrödinger equation. To solve the two-way coupled system, the
Schrödinger equation and Poisson’s equation are solved iteratively until a
self-consistent solution is obtained. The iterative procedure consists of the
following steps:
Step 1
To provide a good initial condition for the iterations, we solve Poisson’s
equation
(1)
for the electric potential,
space charge density.
, in which
is the permittivity and
is the
In this initialization step,
is given by the best initial estimate from physical
arguments; for example, using the Thomas-Fermi approximation.
Step 2
The electric potential, , from the previous step contributes to the potential
energy term,
, in the Schrödinger equation
(2)
where
is the charge of the carrier particle, which is given by
(3)
where
is the charge number and
is the elementary charge.
Step 3
With the updated potential energy term given by Eq. 2, the Schrödinger
equation is solved, producing a set of eigenenergies,
corresponding set of normalized wave functions,
.
Step 4
, and a
The particle density profile,
, is computed using a statistically weighted
sum of the probability densities
(4)
where the weight,
, is given by integrating the Fermi-Dirac distribution
for the out-of-plane continuum states (thus depending on the spatial
dimension of the model).
(5)
(6)
(7)
where
is the valley degeneracy factor,
is the Fermi level,
is the
Boltzmann constant,
is the absolute temperature,
is the density of
state effective mass, and
and
are Fermi-Dirac integrals.
For simplicity, the weighted sum in Eq. 4 shows only one index, , for the
summation. There can be, of course, more than one index in the summation.
For example, in the nanowire model discussed here, the summation is over
both the azimuthal quantum number and the eigenenergy levels (for each
azimuthal quantum number).
Step 5
Given the particle density profile,
, we reestimate the space charge
density,
, and then re-solve Poisson’s equation to obtain a new electric
potential profile, . The straightforward formula for the new space charge
density
(8)
almost always leads to divergence of the iterations. A much better estimate
is given by
(9)
where
is the electric potential from the previous iteration and
additional tuning parameter. Eq. 9 is used by the solver sequence to
compute the space charge density, .
is an
The formula is motivated by the observation that the particle density,
,
is the result from
and would change once Poisson’s equation is resolved to obtain a new . In other words, Eq. 8 can be written more
explicitly as
(10)
since
is the result from
equation to get a new .
, and
is used to re-solve Poisson’s
To achieve a self-consistent solution, a better formula would be
(11)
At this point,
is unknown to us, since it comes from the solution to
the Schrödinger equation in the next iteration. However, we can formulate a
prediction for it using Boltzmann statistics, which provides a simple
exponential relation between the potential energy,
particle density,
.
, and the
(12)
This leads to Eq. 9 for the case of
. This works well at high
temperatures, where Boltzmann statistics is a good approximation. At lower
temperatures, setting
to a positive number helps accelerate convergence.
Step 6
Once a new electric potential profile, , is obtained by re-solving Poisson’s
equation, compare it with the electric potential from the previous
iteration,
. If the two profiles agree within the desired tolerance, then
self-consistency is achieved; otherwise, go to step 2 to continue the iteration.
A dedicated Schrödinger-Poisson study type is available to automatically generate
the steps outlined above in the solver sequence.