SEMITIP V6, Uni2, Example 6: undoped GaAs(110), with on-center point charge
Click here for source and input/output files for Example 6
This example illustrates the potential from a point charge, extended over a radius of 1 nm and centered around r=0. The charge is defined by modifying the RHOSURF routine by adding the statements
IF (R.LT.RADQ) RHO=-1.E14/(PI*RADQ**2)
The probe tip is placed a relatively large distance, 10 nm, from the surface so that it only weakly affects the potential at the surface. Examing the potential profiles in FORT.11 and FORT.12 (green lines), we find a result that agrees reasonably well with the expected 1/sqrt(r2+z2) fall-off (as shown by the red dashed lines) for r or z greater than the 1 nm radius of the surface charge:
The magnitude of the computed and expected potential curves is also in agreement: for the radius on the surface of 1 nm and charge density of -1014/πr2 cm-2=-1/πr2 nm-2 where r is measured in nm,
the total charge is -e so that the expected potential variation as shown by the red dashed lines is 2e2/[(ε+1) sqrt(r2+z2)] = 0.208 eV nm/sqrt(r2+z2) with a dielectric constant of ε=12.9 from the FORT.9 input file.
This agreement between the computed potential and the known result provides an important test of the validity of the details of our finite-difference method, particularly for the revised method employed in SEMITIP version 6 for
computation of derivatives. In contrast, results for the computed potential using the method of version 4 or 5 are shown by the blue dashed and dotted lines. The former gives the absolute magnitude of the computed potential, which is significantly below the expected curves (red). If we scale that blue dashed lines by a factor of 1.5 then we arrive at the blue dotted lines which illustrate that, even aside from the error in magnitude, the fall-off with increasing r or z of these version 4 or 5 results is too fast compared to the theoretical expectation. The approximate agreement between the version 6 results and the theoretical expectation holds over distances of about 1 - 10 nm. Beyond that, the fact that Dirichelet boundary conditions are used, i.e. zero potential at the boundaries, produces a computational results that is significantly less than the theoretical expectation (although they are both quite small, so this disagreement would only be relevant in very specialized types of measurements).