SciPost Submission Page
Electrostatics in semiconducting devices II : Solving the Helmholtz equation
by Antonio Lacerda-Santos, Xavier Waintal
Submission summary
| Authors (as registered SciPost users): | Antonio Lacerda-Santos |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2507.03131v1 (pdf) |
| Date submitted: | July 19, 2025, 3:24 p.m. |
| Submitted by: | Antonio Lacerda-Santos |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
|
| Approaches: | Theoretical, Computational |
Abstract
The convergence of iterative schemes to achieve self-consistency in mean field problems such as the Schr\"odinger-Poisson equation is notoriously capricious. It is particularly difficult in regimes where the non-linearities are strong such as when an electron gas in partially depleted or in presence of a large magnetic field. Here, we address this problem by mapping the self-consistent quantum-electrostatic problem onto a Non-Linear Helmoltz (NLH) equation at the cost of a small error. The NLH equation is a generalization of the Thomas-Fermi approximation. We show that one can build iterative schemes that are provably convergent by constructing a convex functional whose minimum is the seeked solution of the NLH problem. In a second step, the approximation is lifted and the exact solution of the initial problem found by iteratively updating the NLH problem until convergence. We show empirically that convergence is achieved in a handfull, typically one or two, iterations. Our set of algorithms provide a robust, precise and fast scheme for studying the effect of electrostatics in quantum nanoelectronic devices.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
This paper describes a clever way to numerically handle the self-consistent quantum electrostatics (SCQE) problem by mapping the original problem onto a non-linear Helmholtz(NLH) equation. The key is to replace the integrated local density of states at each position by a much simpler piecewise linear function that is locally refined at each iteration. The solution of this approximate problem converges to that of the SCQE as the local density of states is locally refined. The authors show that gradient descent approach must arrive at a unique solution.
This approach seems quite useful for solving non-local problems with non-linearities. I have a couple of questions: 1. Why can NLH not overshoot during an iteration? (Page 10) 2. Do the authors have insight on when the Newton-Raphson scheme is expected to fail or succeed? (Page 9 and Figure 9)
Issues: 1. NLH and NHL are used interchangeably throughout the paper 2. Trivial notes (spell check may be helpful) 1. Redondant -> Redundant 2. Eponym -> eponymous 3. semi-definite positive -> positive semi-definite
Recommendation
Publish (meets expectations and criteria for this Journal)
Report
In this second part of their series of articles about electrostatics in nanostructures, the Authors describe a robust recipe to solve the Schrödinger-Poisson problem, and showcase its application to the case of a nanowire with hexagonal cross-section.
On the technical level, the research presented here represents a remarkable advance. Furthremore, the article is very clearly written. Given that this type of calculations are becoming more and more important and widespread, I can expect that the method proposed will be of high impact.
I have some minor objections related to clarity:
1) The relation with the PESCA method presented in the first installment of the series is not made very clear. In fact, I think that a reader interested in using the method to tackle a Schrödinger-Poisson problem may skip the first article and study directly this second one. On one hand, this is good since it makes this article self-contained. On the other, it would be better to explicitate the relation between the first and second article of the series. (Note: this comment of mine mirrors what I wrote in my report for the first article).
2) In Section 5.2 the Authors say their simulations has "5320 sites in total including the 161 quantum sites which are treated self-consistently". I assume that the 161 sites are those shown explicitly in e.g. Figure 6, forming the hexagonal semiconducting wire cross-section. Where are all the other ~5000 sites and why so many are needed? I hope the Authors can provide more details on this aspect of the simulation, as an opportunity to also explain some criteria on how to choose the discretization grid in practice. It is often this nitty-gritty details that make a method successful or not and so, even if general receipre cannot be provided, examples help.
3) The Authors mention several times that the electrostatic problem is non-local. However, in the discretized version described in Sec. 2, the only non-zero matric elements of the capacitance matrix are diagonal ones and nearest-neighbor ones, since it is essentially a discretized Laplacian. This comes from the first-order approcimation of the gradiant. I suppose it is this feature that makes the problem convergent, as opposed to the full SCQE problem? If this is so, I would emphasize this point.
Recommendation
Ask for minor revision
Report
The manuscript is well written and provides a clear exposition of prior work and of the present contribution. While I find the work sound and of potential interest to the community, I encourage the authors to clarify a few points. In particular, it would be useful to more explicitly contrast their approach with prior nonlinear Poisson or Newton–Raphson schemes, to provide quantitative benchmarks, and to discuss practical limits of the method. Some additional algorithmic details (e.g. pseudocode or a flowchart of the self-consistency loop) would make the implementation more reproducible. Finally, short outlook on possible extensions (e.g. more complex geometries or disorder) would strengthen the paper.
Minor comment: on page 9, the notation for the energy should read
$E^{\alpha(i)}_i$ not $E^{\alpha}_i(i)$. Please correct this typo.
Overall, this is a strong and well-presented manuscript. I recommend publication after minor clarifications and improvements in presentation.
Recommendation
Ask for minor revision