Localization Length vs Disorder

I've done some more studies with time evolution, and placed the results in a publicly accessible google drive directory linked below. In the "disorder" folder is a collection of plots that show the time evolution of the one electron system across a range of "disorder" values (the potential is defined as E_0 + R, where R is a random number from a gaussian distribution with mean of zero and standard deviation of "disorder"). In the moving_I/2 folder are a collection of similar plots, but now with a sloping potential defined as E_0 + R + well*(I/2).

Here is an interesting plot. This shows something we could call the localization length, as a function of disorder. It is calculated with zero sloping potential but with a site disorder as outlined above. The horizontal access is disorder. The quantity on the vertical access is the large-time asymptotic value of the width of the wave-function. Like,  as a function of time, the wave-function gets wider for a while but then it stops getting wider and settles on a particular width, hence we can call that a localization length. (For zero disorder, that would be infinite because it never stops getting wider not matter how long you go int time.)

I am not sure actually how this is defined since a wave-function width less than 1 is a little puzzling. Maybe there is a normalization issue or something? Also, another point is probably to get really accurate values at low disorder one might need to let the wave function evolve for a longer time. I wonder if there is a threshold level of disorder at which this quantity changes from being infinite (no localization) to finite?



It seems that (according to simTest40_exp.gif, which shows the expectation value vs time) at the levels of disorder I've provided there is almost no difference in the speed that the wave moves, regardless of disorder level. For that matter, it appears that it almost immediately becomes a linear movement, and does not (as I had first assumed) accelerate. Does anybody have any idea as to what slope values I might test (the current slope being I/2)?

Here is the link to the drive:  Time Evolution Plots

*EDIT: I've done more simulations, now using a slope of I/2000 and I/200 instead of I/2. They can be found in moving_I/200 and moving_I/2000 respectively. I've also added more plots to the disorder folder (simTest31...). The new plots are also now in both a gif form and an mp4 form (so you can pause, rewind, etc).