Unambiguous Simulation of Diffusive Charge Transport in Disordered Nanoribbons

Very detailed and relatively easy to follow;

In depth analysis of the onset of the scale invariant diffusive regime;

More in the text.

The discussion of the role of the ratio xi/L (localization length/sample size) for the diffusion (Fig.8) is not sufficiently clarified, cf. the text of the report;

In my opinion, both panels of Fig.9 cannot be compared directly: the relation between the both abscissas is not clear at all;

More in the text.

Dear Editor,

You have asked my opinion on the sent in manuscript authored by Veiga, João, Pinho, Pires, and Parente Lopes, which concerns some aspects of the physics of diffusive charge transport in disordered nanoribbons. According to the authors themselves, the main achievement of their contribution with the approximate numerical approach to the transport in 2D systems proposed in it, is the possibility to controllable distinguish between three expected transport regimes, namely ballistic, diffusive and localized.

The manuscript at it is represents an extended version of the Master of Science thesis of the first author, which can be found in the library of the University of Porto. For this reason it is written in a very detailed manner and can be recommended for publishing simply because of its pedagogic value: An interested reader will find in it a thorough introduction to this research field and a sufficiently well described application manual for the employed techniques.

The main attention in the original part is devoted to the transport studies of 2D nanoribbons in framework of the Landauer formalism. The model presented in Section 2.1 seems to be sufficiently realistic. In particular it contains the disorder in form of the onsite random potential with a box distribution.

The application of the Landauer formalism to 2D systems is the matter of a discussion lasting already for several decades with personal opinions ranging from total denial of any predictability
to absolute acceptance. Clear is that while the formalism works very well for 1D case, the application in 2D case relies on approximations.

These are introduced and discussed in and around Section 3 and tried on a 1D toy model with
subsequent extension to a quasi-2D systems. While the results for 1D model presented graphically in Fig.5 seem to point to a consistent picture, with the numerical instabilities showing themselves only at very large times, it is still not quite clear how good or bad do these results compare to the known ones. I presume that there should be a vast knowledge accumulated in the field, so a comparison to some sources would be very good.

In Section 4 the emergence of the diffusive regime is discussed. The results are presented in Fig.8, which shows the dimensionless conductance of the quasi-2D disordered ribbon as function of the system size for fixed disorder strength and three different Fermi energies. The onset of the diffusive regime is observed only at half-filling, i.e. for the Fermi energy adjusted to zero and importantly either in thermodynamic limit (system size to infinity), or in the regime with infinite localization length xi, cf. Fig 8c. I would expect a more thorough discussion of this point. For instance, is it possible to model a finite size system with the disorder chosen such that the localization length xi is much larger than the system size? Would one then see the onset of the diffusive regime?

The authors argue that the ultimate test for the correctness of their approximations and approach schemes comes from the comparison with the results obtained from the static Kubo-Greenwood formula in Eq.(49) and shown in Fig.9. Although their talk here about a "direct comparison", I have my difficulties with linking the respective lengths to each other. The Kubo-Greenwood formula is not subtle with respect to the spatial dimension of the system, the trace simply becomes the integral over the Brillouin zone. What is the relation between the respective lengths and how such length appears in the case of the Kubo-Greenwood formula if the evaluation is performed over an infinite sample? At half filling the system lacks any specific intrinsic length.

To conclude: Although I generally tend to take a positive stand towards the publication recommendation of this nice work, I still have some of the reservation. If the authors respond satisfactory to my objections I would be happy to do just so.

In the text

Ask for minor revision

1- A new method to compute quantum transport in 2D materials. This makes a positive contribution to a currently very active field of research into the electronic properties of nanomaterials.
2- The manuscript is well written and the numerical method is well presented.

1- Perhaps the link with experimental results could be more detailed and explicit in the manuscript.

This manuscript proposes a new numerical method for calculating the conductance in a 2D systems in the presence of an Anderson defects. It is an extension to a ribbon of variable width of the work proposed for a 1D chain. in PRB 101, 104203 (2020) Ref. [68]. That method allows them to simulate the diffusive transport regime expected in 2D systems of size between the mean-free path and localization length.
The work is well presented and appropriate in terms of the description of the numerical method used and the quality of the calculations carried out. It seems to me that these results are new and interesting and deserve to be published. However, I have some comments/questions that the authors could take into account to improve the manuscript.

1- Why this new method? What is really new or advantageous, that cannot be calculated using other methods? In particular those, method based on Kubo-Greenwood approach described in reviews Refs.13 and 25 and in other references cited in the manuscript. It seems to me that other numerical methods can also capture the diffusivity plateau in 2D systems.
Moreover, I am surprised by the results shown in figure 9(b) obtained in a Kubo-Greenwood method which is not detailed in this manuscript. The authors conclude that their method does a better job of capturing the effect of geometry, but can they comment a little on why Kubo-Greenwood calculation does not work.
2- The static disorder studied is of the Anderson disorder type. This is entirely appropriate for simulating the effect of a random distribution of non-resonant scatterers. In 2D materials, resonant scatterers such as vacancies or adsorbed atoms are also very important from an applications point of view. Can the method used be applied in this case?
3- The last sentence of section 2.2 is: “From here on forward, the presented timedependent results were computed with f(t) = 1 − e(− τt), where τ is an adiabatic parameter.” Can you explain what this adiabatic parameter is and how it affects the results?

Publish (meets expectations and criteria for this Journal)