Reconstructing the potential configuration in a high-mobility semiconductor heterostructure with scanning gate microscopy

1-High innovative potential with a substantial interdisciplinary component

2-Clear concept

3-May become important for device fabrication

Since this is a quite new approach, I would not rate still existent open questions as really weak points. The paper more likely points out a way towards establishing a new characterization method that, of course, has still room for improvements (see report). This kind of weakness may also be rated as a strength by possibly opening up a new field for the community to deal with.

The Authors aim at using SGM maps that consist of the measured conductance response of an electronic system depending on the lateral position of an SGM tip. Simply speaking, the SGM tip probes the local properties of the electron system that finally should provide information of the underlying lateral distribution of the random disorder potential.

Because of the ongoing minimization of electronic devises the development of such a method for determination of the individual fingerprint of potential fluctuations is of major importance. In the case that the size of the devices get of the order of the size of individual fluctuations, a pure statistical representation of disorder may no longer be correct.

Although there can be expected a causal relation between a particular lateral disorder potential distribution and a particular lateral conductance response pattern caused by the SGM tip position, it is hopeless to look for a physical model that directly maps lateral conductance patterns onto the lateral disorder potential distribution. Therefore the authors propose and apply machine learning to overcome that challenge.

However that challenge becomes even bigger because of the fact that there are no experimental training data available for the learning algorithm, which would require to have that problem already solved for at least some particular real cases.

For the training of the learning algorithm the Authors therefore use simulated transport data that result from particularly designed artificial (known) disorder potentials. For getting the training data they have to manage two major tasks: While task one consist of modelling the position dependent impact of the SGM tip on the electronic system by solving the Schrödinger – Poisson problem self consistently, the second task is generating the transport data. The latter is based on the widely used software package KWANT, while the first task is laid out in detail in in previous work of some of the authors.

On the background of the above very briefly summarized concept I can say that the paper is innovative, clearly written and the description of the different tasks is scientifically sound.

Already at this point I would highly recommend publication, although there still can be found reasons for criticism, as it is mostly the case if introducing new methods with high innovative potential. However, we should not forget that one of the major missions of a publisher is to contribute to a constructive discourse within the scientific community and at least to some extend the readers should get their own chance for criticism. Therefore still open questions should not prevent publication in the first place, as long as all ingredients are laid out correctly.

At this point I would like to address one of those possibly still open questions on my own. In task one the authors address screening on the Thomas – Fermi and/or Hartree level based on previous work. In a bench mark test in that previous work they address the IQHE problem and end up with results for the potential and carrier distribution at the edge that agree very well with a famous and highly cited publication of D. B. Chklovskii, B. I. Shklovskii and L. I. Glazman (Ref.36). That gives the authors confidence that they address the electronic system correctly. However, about 10 years ago Pascher et al (DOI10.1103/PhysRevX.4.011014) addressed the edge channel transmission of QPCs in the QHE regime via SGM and found out that they cannot successfully model their transmission data based on a modelling relying also on Thomas-Fermi screening. They argue that in such a model screening might be considerably over estimated. Some most recent citations of that Pascher paper seem to challenge the picture of Chklovskii, Shklovskii and Glazman, which finally might have an impact also on this paper.

Since machine learning has nothing to do with the underlying physics, training with data from a physically wrong model would always allow to regain the correct underlying artificial test potential. The application to a real experimental SGM image then would also result in a particular disorder potential. However, that resulting image might still not have much to do with the real potential because of looking at it with the “wrong eyes”.

However, there is a point that may be in favour of the author’s model, namely the fact that the QHE regime uses of course high magnetic fields that might modify interactions considerably. The authors in this paper use their model without magnetic field, which might justify the Thomas-Fermi approach to be a sufficient approximation. Maybe the Authors want to add a comment on this?

Anyway, I think the Authors did the best that could be done at this stage. They even tried some sort of modifying the real experimental conditions (such as e.g. different openings of the QPCs) to eliminate side effects, since these should not affect the predicted potentials. As I already said, despite the fact that there still can be found open questions, the paper should be made public in order to fuel the discussion within the scientific community and stimulate further developments.

On this background I even tend to recommend publication as it is, unless the Authors take the option of making a comment as mentioned above, which, of course, is not obligatory from my side!

Like stated above, there is room for improvements of the method and to some degree that should also come from the scientific community. Therefore I find it much more important to publish as soon as possible instead of digging out further things that probably still could have been done in the paper.

Therefore no changes required from my side.

1. It presents an approach for estimating the disorder background potential of quantum constrictions based on scanning gate measurements and a machine learning approach.
2. It presents details of the computations that allows the simulations to be reproduced.
3. It offers a comprehensive discussion about the techniques and approaches used.

1. It only uses one machine learning technique. There may be techniques that offer better results, but this topic is not explored.
2. The technique of transfer learning is used. Although, it is probably the best that can be done, it offers no guarantees that the estimated potential are close to those one would obtain experimentally.

This is a timely contribution to the field and shows details of an approach for estimating the disorder potential experienced by electrons in a quantum constriction. This estimation is particularly important for designing quantum effect devices, where small variations in the disorder potential can lead to drastic changes in the properties of the device. I recommend the publication of this work and propose a few additions to the paper to make it stronger.

1. Hafnium oxide has piezoelectric properties. A tip scanning its surface may induce a piezoelectric polarization of the layer, which may affect the background potential. The authors could add a brief discussion if this is really the case, and how much the potential could be affected.
2. In page 4, the authors say that the "Fermi wavelength ... has been experimentally determined". How was it experimentally determined?
3. What kind of calculations does KWANT do? In this sense, it would be useful to know where the edge of the first Brillouin zone is compared to the Fermi wavelength. This can lead to limitations in several computational techniques such as tight-binding.
4. The calculations are performed for a single electron. Therefore, the estimated potentials should correspond to potentials that a single electron would experience. However, it is not made explicit in the discussions if the transport is actually a result of a single electron transport. It would be helpful to add some discussion that clarifies this point.
5. Also, related to the previous point. It would be helpful to the reader if the authors provided some comparison to potentials experimentally measured elsewhere. Are the estimated potentials similar to the experimentally expected potentials?
6. Another point related to the previous: After reaching the end of the paper, I had a feeling that I could have learned something from the estimated potentials. What can be extracted from this information? Does it help in designing new devices? How? etc. What does one gain knowing the potential profile of the background?
7. In page 11 the authors say: "data already yields good results". What does good mean? A qualitative description would be more useful.
8. It would be very helpful to the reader to see a plot of the quality of training as a function of the number of training samples. This would convey a broader idea about the limitations of the technique. For example, it would indicate if the technique could be improved with a larger training set.

Gaëtan et al. present a study wherein they utilize an encoder-decoder neural network (NN) to infer potential configurations from experimental scanning gate microscopy (SGM) data. Their approach integrates PTFS self-consistent calculations and KWANT to generate SGM maps and consequently, the training data for the NNs. An intriguing improvement is the adoption of transfer learning, where they initiate with a PLDOS pre-trained NN. This reportedly overcomes challenges related to data acquisition for training.

In terms of validation, the authors change only the QPC gate voltage, maintaining other parameters unchanged. While this results in changes in the SGM images, the predicted potentials remain largely invariant, providing indirect evidence of their method's physical robustness. An interesting observation they make is the beneficial effect of applying a spatial Gaussian filter to the data, which seemingly enhances reliability.

The findings could potentially pave the way for numerous scanning microscopy experiments to extract spatially resolved local physical quantities using similar techniques. Given its novelty, I believe this work is suitable for publication in SciPost Physics, contingent on addressing the below concern:

Regarding the inverse problem-solving from potentials to SGM measurements through the NN: How well does this method maintain locality? For instance, if Fig.2 (left) were confined to a 100nm x 100nm region, how would both the simulation and NN predictions be affected? A test of this nature would provide valuable insights into how the NN works.

1. A general discussion about what physically quantity that SGM is measuring can help improve the readability.
2. In general, I think SciPost does not have a mandatory request for code avalibility. But I think open-source sample code and model can help reader better understand the achievements that the authors made, and increase the reproducibility. Thus, I kindly request the open-source code/data of this study.