# TranSiesta Hartree correction for FFT Poisson solver

TranSiesta calculations based on the NEGF formalism is based on solving the Poisson equation for an open-boundary condition. For 2 electrodes (left/right) a potential drop between the electrodes.

In Siesta the Poisson equation is solved using Fourier transforms which implicitly assumes periodic boundary conditions. This is contrary to the actual open boundary conditions which have fixed and different potentials at the boundaries.

In 2-probe calculations this is trivially fixed by adding a ramp to the unit-cell along the transport direction. This fixes the periodic boundary conditions.

General $$N$$ electrode calculation cannot easily fix the periodic boundary conditions and this forces the users to intervene. The fdf option TS.Poisson should be used to supply a correction term to the Poisson solution that enforces the boundary conditions imposed by the electrodes. However, creating such a correction is not trivial.

This tool tries to remedy this by solving the Poisson equation with fixed boundaries as defined by the user.

## Command line tool

It may be called using

stoolbox ts-fft --help


which gives a lot of instructions.

Here we list the required items needed to create a correction file compatible with TranSiesta.

1. Supply a geometry from a previous TBtrans calculation

2. Manually define the applied bias’ for each electrode

3. Define the shape of the grid, use values from Siesta output; search for line InitMesh: MESH = A x B x C

4. Define boundary conditions for each of the simulation box 6 sides

5. Define the output file

All of these points are very important, while users should in particular take note of point 4.

An example command line for a bulk system with transport along third lattice vector would look something like:

stoolbox ts-fft --geometry siesta.TBT.nc -V Left 0.5 -V Right -0.5 \
--shape 300 200 100 --out fft-fix.TSV.nc \
--boundary-condition-c d d


Note the boundary conditions along the transport direction; here d means Dirichlet.

For systems where there is no periodicity one can use Neumann boundaries. For a 1D chain one would do:

stoolbox ts-fft --geometry siesta.TBT.nc -V Left 0.5 -V Right -0.5 \
--shape 300 200 100 --out fft-fix.TSV.nc \
--boundary-condition-a n n \
--boundary-condition-b n n \
--boundary-condition-c d d


Which uses Dirichlet for transport direction, otherwise Neumann.

### Better performance

Sometimes the grid used is very large --shape. In these case it may be beneficial to solve the Poisson equation for a smaller shape, and then interpolate. This is accomblished using both --pyamg-shape (Poisson solution) and --shape (output shape). An interpolation will be done after the solution step.

## Method

The internal method solves the Poisson equation twice using a multigrid solver (pyamg). First it fixes the potentials on all grid points touching the electrode atoms (atomic radius controlled with --radius). Then uses the solution to find all boundary point values, fixes these and solves the Poisson equation again. The resulting solution should somewhat match the boundary conditions of the calculation and correct the FFT solution.

If you know the boundary conditions that fixes the FFT solution for your particular setup, you are encouraged to use that instead.