Comparison of methods to minimize the electronic total energy

This is for insulator and semiconductor systems only

Go here for metals

SYSTEMS

GaAs displaced

All methods work well.

SCF methods

PK – Pulay_kerker mixing ; PTF – Pulay-Thomas-Fermi mixing ; Br – Broyden

Dir – direct minimization methods. i.e. no potential mixing

Minimization methods: MG – Mauri Galli functional ; GCG – Grassmann algorithm

The Broyden swhows its inability. “tf.gcg” does have some problem. The difference between the other methods is imperceptible

Again we have a tie between “tf.mg”,“dir.gcg”, and “PK mg”. “tf.gcg” again is a bit slower. The Broyden method again is the worse. There are some jumps in the timing of the “ dir gcg” method. These are due to the job being swapped out of the CPU.

GaInAs

We look at unit cells extended in the (110) direction of GaAs stacked on top of InAs. We use 6,10, and 20 layers of GaAs on top of respectively 6,10, and 20 layers of InAs. A 1x4x4 Monkhorst-Pack mesh is used for the k-point sampling.

For the 6-layer system, we can see that all methods perform fairly well. The Broyden method “ br” shows early deficiencies. The “krk” signifies that the Pulay_Kerker mixing is used: “tf” Pulay_Thomas-Fermi and “dir” signifies the direct minimization method. “mg” signifies the Mauri-Galli functional is used for the electronic minimization and “gcg” signifies the Grassmann conjugate gradient algorithm.

The convergence of the Broyden method shows worsening. There is not a significant difference between the other methods. The “dir.gcg” and the “tf.mg” methods appear to be the best, but the Pulay-Kerker and Pulay-Thomas-Fermi mixng with the GCG minization do not differ significantly.

Again the “dir.gcg” and the “tf.mg” methods converge at almost the exact same time. We have dropped the Broyden scheme as it had already proven to be insufficient in the 10-layer system.

GaAs surface

Similarly we used unit cells of 6,10, and 20 layers of GaAs and 6,10, and 20 layers of vacuum creating surfaces in the (110) direction. The same k-point sampling is used.

Again for the small system all methods handle it pretty well. The “tf.mg” setting shows a slight instability at the end.

At this point we really see a large benefit of the Thomas-Fermi mixing. The “tf.gcg” and the “dir.gcg” perform almost exactly the same. We can see an instability in the “tf.mg” method. At the point this method strays from the “tf.gcg” method, the “tf.mg” has a dip to a unphysical lower energy before settling into the higher proper value. The absolute value of the energy difference is plotted for this set of data. The dip may be because in the total energy expression we use the band energy obtained from the input potential and not of the output potential. We do include a correction, but it is only a first-order correction. The Pulay_kerker method begins to show weakened convergence. The Broyden method performed very poorly for this system.

Now the instability is in the “tf.gcg” alorithm. It has a dip in energy as the”tf.mg” did in the 10-layer system. Not a substantial difference between al three, but “tf.mg” does the best. The Pulay-Kerker and the Broyden method were dismissed for this system due to their poor performance in the 10 layer system.

GaAs primtive unit cell

For a few k-points, the direct method and the SCF method are about equal.

For many k-points, we see an appreciable difference in these methods. We are not sure exactly the cause of this. We believe it to be inherent in the direct method.