pith. sign in

arxiv: 2507.03131 · v5 · submitted 2025-07-03 · ❄️ cond-mat.mes-hall · physics.comp-ph

Electrostatics in semiconducting devices II: Solving the Helmholtz equation

Antonio Lacerda-Santos , Xavier Waintal This is my paper

Pith reviewed 2026-05-19 05:44 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall physics.comp-ph
keywords self-consistent electrostaticsquantum nanoelectronicsnon-linear Helmholtz equationconvex functionalSchrödinger-Poissoniterative methodsThomas-Fermi approximation
0
0 comments X

The pith

Mapping the self-consistent electrostatic problem to a non-linear Helmholtz equation permits provably convergent iterations that reach the exact solution in one or two steps.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the unreliable convergence of standard iterations for self-consistent quantum-electrostatic problems such as the Schrödinger-Poisson equation, especially when nonlinearities become strong. It maps the original problem onto a Non-Linear Helmholtz equation at the price of a controllable error. For this approximated equation a convex functional is constructed whose minimum coincides with the desired solution, so that standard minimization algorithms are guaranteed to converge. The approximation is then lifted by repeatedly updating the Helmholtz problem with the latest density, and the exact solution is recovered after a small number of such outer iterations.

Core claim

We map the self-consistent quantum-electrostatic problem onto a Non-Linear Helmholtz equation, a generalization of the Thomas-Fermi approximation. We construct a convex functional whose minimum is the sought solution of the NLH problem and thereby obtain iterative schemes that are provably convergent. The approximation is subsequently lifted by iteratively updating the NLH problem until the exact solution of the original problem is reached, with convergence typically occurring in one or two iterations.

What carries the argument

The Non-Linear Helmholtz equation together with an associated convex functional whose global minimum supplies the unique solution of the approximated problem.

If this is right

  • Standard iterative schemes for Schrödinger-Poisson problems become reliable even when the electron gas is partially depleted or subjected to large magnetic fields.
  • The method supplies a precise and fast computational tool for electrostatics inside quantum nanoelectronic devices.
  • Convergence occurs after only a handful of iterations rather than exhibiting capricious behavior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The separation between an inner convergent solver and an outer lifting loop could be applied to other mean-field problems whose direct iteration is unstable.
  • The convex-functional construction may lend itself to proofs of global convergence rates or to parallel implementations.
  • Because the inner problem is now convex, existing optimization libraries can be reused without custom tuning for each device geometry.

Load-bearing premise

The original self-consistent problem can be replaced by the Non-Linear Helmholtz equation with an error small enough that a few outer iterations remove it completely.

What would settle it

A numerical experiment on a device with strong depletion or large magnetic field in which the outer lifting iterations either diverge or fail to reproduce the solution obtained by a well-converged reference solver.

Figures

Figures reproduced from arXiv: 2507.03131 by Antonio Lacerda-Santos, Xavier Waintal.

Figure 1
Figure 1. Figure 1: Construction of the piecewise linear ILDOS [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Solving the NLH equation using the piecewise linear Helmholtz algorithm: [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Solving the NLH equation using the piecewise linear Helmholtz algorithm: [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Solving the NLH equation using the piecewise linear Helmholtz algorithm: [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Error on the chemical potential and charge (max over all sites) as a function [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Solving the NLH equation using the piecewise linear Helmholtz algorithm: [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Local density of states (green) and Integrated local density of states (black) [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Charge profile for the self-consistent problem defined in Eq. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Convergence of the SCQE loop for Vbg of −0.02V (green), −0.05V (red) and −0.06V (black). Different symbols correspond to Piecewise Linear Helmholtz algorithm (crosses) and Piecewise Newton Raphson algorithm (circles). We obtain machine precision for Piecewise Linear Helmholtz while the error of the Piecewise Newton Raphson calculation stagnates for −0.05V (red) and −0.06V (black). 7 Conclusion In this pape… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that the self-consistent Schrödinger-Poisson problem in semiconducting devices can be mapped to a Non-Linear Helmholtz (NLH) equation (a Thomas-Fermi-like surrogate) at the cost of a small error. It constructs a convex functional whose minimum yields the NLH solution, enabling provably convergent inner iterations. An outer lifting procedure then iteratively updates the NLH source term until the exact quantum-electrostatic solution is recovered, with empirical evidence that convergence occurs in typically one or two iterations. The resulting algorithms are presented as robust, precise, and fast for studying electrostatics in quantum nanoelectronic devices under strong nonlinearity.

Significance. If the mapping error remains small and controllable and the lifting loop converges rapidly as claimed, the approach would address a persistent practical difficulty in mean-field quantum-electrostatic calculations, particularly in partially depleted regimes or strong magnetic fields. The explicit use of a convex functional to guarantee inner-loop convergence is a methodological strength that could be adopted more broadly. The reported empirical efficiency (one or two outer iterations) suggests immediate computational utility, provided the theoretical control on the approximation error is supplied.

major comments (2)
  1. [Abstract] Abstract: the central claim that the self-consistent quantum-electrostatic problem maps to the NLH equation 'at the cost of a small error' that is subsequently removed by lifting lacks any a priori bound or scaling estimate on the mapping error (e.g., with depletion depth, magnetic length, or temperature). Without such control, the assertion that the outer iteration terminates in one or two steps cannot be extrapolated beyond the tested cases and is load-bearing for the overall method.
  2. [Abstract] Abstract: the convex functional whose minimum is stated to be the NLH solution is invoked to prove convergence of the inner iteration, yet neither its explicit mathematical form, the proof of convexity, nor the precise lifting update rule for the source term is supplied. These omissions prevent verification of the 'provably convergent' property and constitute a load-bearing gap for the central algorithmic contribution.
minor comments (2)
  1. Typos: 'seeked' should read 'sought'; 'handfull' should read 'handful'.
  2. The abstract refers to 'Our set of algorithms' without enumerating or distinguishing them; a brief listing or reference to the relevant sections would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. The comments identify important gaps in the presentation of theoretical controls and algorithmic details. We have revised the manuscript to address both major comments by adding the requested a priori estimates and the explicit mathematical constructions, while preserving the empirical focus of the original submission.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the self-consistent quantum-electrostatic problem maps to the NLH equation 'at the cost of a small error' that is subsequently removed by lifting lacks any a priori bound or scaling estimate on the mapping error (e.g., with depletion depth, magnetic length, or temperature). Without such control, the assertion that the outer iteration terminates in one or two steps cannot be extrapolated beyond the tested cases and is load-bearing for the overall method.

    Authors: We agree that an a priori bound would strengthen the claim. In the revised manuscript we have added a new subsection deriving scaling estimates for the mapping error in terms of depletion depth, magnetic length, and temperature, obtained via a perturbative expansion around the Thomas-Fermi limit. These estimates are corroborated by additional numerical benchmarks that include regimes with strong depletion and large magnetic fields. The outer lifting procedure is shown to converge in one or two iterations whenever the error remains below a controllable threshold, thereby supporting extrapolation beyond the originally tested cases. revision: yes

  2. Referee: [Abstract] Abstract: the convex functional whose minimum is stated to be the NLH solution is invoked to prove convergence of the inner iteration, yet neither its explicit mathematical form, the proof of convexity, nor the precise lifting update rule for the source term is supplied. These omissions prevent verification of the 'provably convergent' property and constitute a load-bearing gap for the central algorithmic contribution.

    Authors: We acknowledge that the original submission omitted the explicit expressions and proof. The revised manuscript now states the convex functional in closed form in the main text, provides a complete proof of convexity in a dedicated appendix (with a concise outline in the main body), and gives the precise source-term update rule used in the outer lifting loop. These additions allow direct verification of the guaranteed convergence of the inner iteration and of the overall algorithm. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on independent variational construction and empirical lifting

full rationale

The paper maps the self-consistent Schrödinger-Poisson problem to an NLH surrogate equation, constructs a convex functional whose minimum is defined to be the NLH solution via standard variational principles, solves the inner problem with a provably convergent iteration, and then applies an outer lifting iteration that updates the NLH source term until the original equations are recovered. The inner convergence is guaranteed by convexity of the constructed functional rather than by any fitted parameter or self-referential definition. The outer-loop speed (one or two iterations) is reported as an empirical observation on tested cases, not as a prediction derived from the inputs by construction. No self-citation chain, ansatz smuggling, or renaming of known results is used to justify the central claims; the approach remains self-contained against external benchmarks such as direct Schrödinger-Poisson solvers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that the NLH mapping introduces only a small, correctable error and that a convex functional exists for the approximated problem.

axioms (1)
  • domain assumption The self-consistent quantum-electrostatic problem can be mapped to a Non-Linear Helmholtz equation with only a small error
    Explicitly stated in the abstract as the starting point for the convergent scheme.
invented entities (1)
  • Non-Linear Helmholtz (NLH) equation no independent evidence
    purpose: Generalization of Thomas-Fermi that permits construction of a convex functional whose minimum solves the approximated electrostatic problem
    Core new object introduced to replace the original self-consistent loop

pith-pipeline@v0.9.0 · 5722 in / 1384 out tokens · 53534 ms · 2026-05-19T05:44:30.937350+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · 2 internal anchors

  1. [1]

    Electrostatics in semiconducting devices I : The Pure Electrostatics Self Consistent Approximation

    A. Lacerda-Santos and X. Waintal, Electrostatics in semiconducting devices i : The pure electrostatics self consistent approximation, doi:10.48550 /arXiv.2502.15897 (2025)

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  2. [2]

    Lacerda-Santos, C

    A. Lacerda-Santos, C. Groth and X. Waintal, Electrostatics in semiconducting devices iii : The pescado open source library, In preparation (2025)

    work page 2025

  3. [3]

    Computational quantum transport: a scattering approach perspective

    X. Waintal, M. Wimmer, A. Akhmerov, C. Groth, B. K. Nikolic, M. Istas, T . Ö. Rosdahl and D. Varjas, Computational quantum transport, arXiv preprint arXiv:2407.16257 (2024), doi:10.48550/arXiv.2407.16257

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2407.16257 2024

  4. [4]

    C. W . Groth, M. Wimmer, A. R. Akhmerov and X. Waintal,Kwant: a software package for quantum transport, New Journal of Physics 16(6), 063065 (2014), doi:10.1088 /1367- 2630/16/6/063065

    work page 2014

  5. [5]

    Chatzikyriakou, J

    E. Chatzikyriakou, J. Wang, L. Mazzella, A. Lacerda-Santos, M. C. d. S. Figueira, A. Trel- lakis, S. Birner, T . Grange, C. Bäuerle and X. Waintal,Unveiling the charge distribution of a gaas-based nanoelectronic device: A large experimental dataset approach, Phys. Rev. Res. 4, 043163 (2022), doi:10.1103 /PhysRevResearch.4.043163

    work page 2022

  6. [6]

    Bautze, C

    T . Bautze, C. Süssmeier, S. Takada, C. Groth, T . Meunier, M. Yamamoto, S. Tarucha, X. Waintal and C. Bäuerle, Theoretical, numerical, and experimental study of a flying qubit electronic interferometer , Phys. Rev. B 89, 125432 (2014), doi:10.1103/PhysRevB.89.125432

    work page doi:10.1103/physrevb.89.125432 2014

  7. [7]

    D. B. Chklovskii, B. I. Shklovskii and L. I. Glazman, Electrostatics of edge channels, Phys. Rev. B46, 4026 (1992), doi:10.1103 /PhysRevB.46.4026

    work page 1992

  8. [8]

    Armagnat and X

    P . Armagnat and X. Waintal, Reconciling edge states with compressible stripes in a ballistic mesoscopic conductor , Journal of Physics: Materials 3(2), 02LT01 (2020), doi:10.1088/2515-7639/ab7582

    work page doi:10.1088/2515-7639/ab7582 2020

  9. [9]

    H. Gummel, A self-consistent iterative scheme for one-dimensional steady state transistor calculations, IEEE Transactions on Electron Devices11(10), 455 (1964), doi:10.1109/T- ED.1964.15364

    work page doi:10.1109/t- 1964

  10. [10]

    I. Tan, G. L. Snider, L. D. Chang and E. L. Hu, A self-consistent solution of Schrödinger–Poisson equations using a nonuniform mesh , Journal of Applied Physics 68(8), 4071 (1990), doi:10.1063 /1.346245, https: //pubs.aip.org/aip/jap/article-pdf/ 68/8/4071/8015524/4071_1_online.pdf

    work page 1990

  11. [11]

    V . Gudmundsson, Oscillating impurity spectra caused by non-linear screening in the quantum hall regime , Solid State Communications 74(2), 63 (1990), doi:https://doi.org/10.1016/0038-1098(90)90606-C

    work page doi:10.1016/0038-1098(90)90606-c 1990

  12. [12]

    L. Wang, D. Wang and P . M. Asbeck,A numerical schrödinger–poisson solver for radially symmetric nanowire core–shell structures, Solid-State Electronics 50(11), 1732 (2006), doi:https://doi.org/10.1016/j.sse.2006.09.013

    work page doi:10.1016/j.sse.2006.09.013 2006

  13. [13]

    F . Rana, S. Tiwari and D. A. Buchanan, Self-consistent modeling of accumulation layers and tunneling currents through very thin oxides , Applied Physics Letters 69(8), 1104 (1996), doi:10.1063 /1.117072, https: //pubs.aip.org/aip/apl/article-pdf/69/8/1104/ 7805404/1104_1_online.pdf. 19 SciPost Physics Submission

    work page 1996

  14. [14]

    F . Stern, Iteration methods for calculating self-consistent fields in semicon- ductor inversion layers , Journal of Computational Physics 6(1), 56 (1970), doi:https://doi.org/10.1016/0021-9991(70)90004-5

    work page doi:10.1016/0021-9991(70)90004-5 1970

  15. [15]

    Trellakis, A

    A. Trellakis, A. T . Galick, A. Pacelli and U. Ravaioli,Iteration scheme for the solution of the two-dimensional Schrödinger-Poisson equations in quantum structures, Journal of Applied Physics 81(12), 7880 (1997), doi:10.1063 /1.365396

    work page 1997

  16. [16]

    V . Eyert, A comparative study on methods for convergence acceleration of iter- ative vector sequences , Journal of Computational Physics 124(2), 271 (1996), doi:https://doi.org/10.1006/jcph.1996.0059

    work page doi:10.1006/jcph.1996.0059 1996

  17. [17]

    A. Vuik, D. Eeltink, A. R. Akhmerov and M. Wimmer, Effects of the electrostatic environ- ment on the majorana nanowire devices, New Journal of Physics 18(3), 033013 (2016), doi:10.1088/1367-2630/18/3/033013

    work page doi:10.1088/1367-2630/18/3/033013 2016

  18. [18]

    D. G. Anderson, Iterative Procedures for Nonlinear Integral Equations, Journal of the ACM 12(4), 547 (1965), doi:10.1145 /321296.321305

    work page arXiv 1965

  19. [19]

    A. E. Antipov, A. Bargerbos, G. W . Winkler, B. Bauer, E. Rossi and R. M. Lutchyn,Effects of gate-induced electric fields on semiconductor majorana nanowires, Phys. Rev. X 8, 031041 (2018), doi:10.1103 /PhysRevX.8.031041

    work page 2018

  20. [20]

    S. D. Escribano, A. Levy Yeyati, Y. Oreg and E. Prada, Effects of the electrostatic en- vironment on superlattice majorana nanowires , Phys. Rev. B 100, 045301 (2019), doi:10.1103/PhysRevB.100.045301

    work page doi:10.1103/physrevb.100.045301 2019

  21. [21]

    Pulay ,Convergence acceleration of iterative sequences

    P . Pulay ,Convergence acceleration of iterative sequences. the case of scf iteration, Chemical Physics Letters 73(2), 393 (1980), doi:https://doi.org/10.1016/0009-2614(80)80396- 4

    work page doi:10.1016/0009-2614(80)80396- 1980

  22. [22]

    C. G. Broyden, A class of methods for solving nonlinear simultaneous equations , Math. Comp. 19, 577 (1965), doi:10.1090 /s0025-5718-1965-0198670-6

    work page 1965

  23. [23]

    Andrei and I

    P . Andrei and I. Mayergoyz, Analysis of fluctuations in semiconductor devices through self-consistent Poisson-Schrödinger computations , Journal of Applied Physics 96(4), 2071 (2004), doi:10.1063 /1.1772886, https: //pubs.aip.org/aip/jap/article-pdf/96/ 4/2071/10643133/2071_1_online.pdf

    work page 2071

  24. [24]

    R. Lake, G. Klimeck, R. C. Bowen, D. Jovanovic, D. Blanks and M. Swaminathan, Quan- tum transport with band-structure and schottky contacts, physica status solidi (b) 204(1), 354 (1997), doi:10.1002 /1521-3951(199711)204:1<354::AID-PSSB354>3.0.CO;2-V

    work page 1997

  25. [25]

    A. Pacelli, Self-consistent solution of the schrodinger equation in semiconductor de- vices by implicit iteration , IEEE Transactions on Electron Devices 44(7), 1169 (1997), doi:10.1109/16.595946

    work page doi:10.1109/16.595946 1997

  26. [26]

    Kumar, S

    A. Kumar, S. E. Laux and F . Stern, Electron states in a gaas quantum dot in a magnetic field, Phys. Rev. B 42, 5166 (1990), doi:10.1103 /PhysRevB.42.5166

    work page 1990

  27. [27]

    A. E. G. Mikkelsen, P . Kotetes, P . Krogstrup and K. Flensberg, Hybridiza- tion at superconductor-semiconductor interfaces , Phys. Rev. X 8, 031040 (2018), doi:10.1103/PhysRevX.8.031040

    work page doi:10.1103/physrevx.8.031040 2018

  28. [28]

    Trellakis and U

    A. Trellakis and U. Ravaioli, Lateral scalability limits of silicon conduction channels, Jour- nal of Applied Physics 86(7), 3911 (1999), doi:10.1063 /1.371307. 20 SciPost Physics Submission

    work page 1999

  29. [29]

    Curatola and G

    G. Curatola and G. Iannaccone, Nanotcad2d: Two-dimensional code for the simula- tion of nanoelectronic devices and structures , Computational Materials Science 28(2), 342 (2003), doi:https://doi.org/10.1016/S0927-0256(03)00117-4, Proceedings of the Symposium on Software Development for Process and Materials Design

    work page doi:10.1016/s0927-0256(03)00117-4 2003

  30. [30]

    Z. Ren, R. Venugopal, S. Goasguen, S. Datta and M. Lundstrom, nanomos 2.5: A two- dimensional simulator for quantum transport in double-gate mosfets , IEEE Transactions on Electron Devices 50(9), 1914 (2003), doi:10.1109 /TED.2003.816524

    work page arXiv 1914

  31. [31]

    Niquet, V .-H

    Y.-M. Niquet, V .-H. Nguyen, F . Triozon, I. Duchemin, O. Nier and D. Rideau, Quan- tum calculations of the carrier mobility: Methodology, Matthiessen’s rule, and com- parison with semi-classical approaches , Journal of Applied Physics 115(5), 054512 (2014), doi:10.1063 /1.4864376, https: //pubs.aip.org/aip/jap/article-pdf/doi/10. 1063/1.4864376/13716499/...

    work page 2014

  32. [32]

    Armagnat, A

    P . Armagnat, A. Lacerda-Santos, B. Rossignol, C. Groth and X. Waintal,The self-consistent quantum-electrostatic problem in strongly non-linear regime , SciPost Physics 7(3), 031 (2019), doi:10.21468 /SciPostPhys.7.3.031

    work page 2019

  33. [33]

    G. W . Winkler, A. E. Antipov, B. van Heck, A. A. Soluyanov, L. I. Glazman, M. Wimmer and R. M. Lutchyn, Unified numerical approach to topological semiconductor-superconductor heterostructures, Phys. Rev. B 99, 245408 (2019), doi:10.1103 /PhysRevB.99.245408

    work page 2019

  34. [34]

    I. M. Flór, A. Lacerda-Santos, G. Fleury , P . Roulleau and X. Waintal,Positioning of edge states in a quantum hall graphene pn junction , Phys. Rev. B 105, L241409 (2022), doi:10.1103/PhysRevB.105.L241409

    work page doi:10.1103/physrevb.105.l241409 2022

  35. [35]

    P . R. Amestoy , I. S. Duff, J. Koster and J.-Y. L’Excellent,A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM Journal on Matrix Analysis and Appli- cations 23(1), 15 (2001), doi:10.1137 /S0895479899358194

    work page 2001

  36. [36]

    P . R. Amestoy , A. Guermouche, J.-Y. L’Excellent and S. Pralet, Hybrid scheduling for the parallel solution of linear systems , Parallel Computing 32(2), 136 (2006), doi:10.1016/j.parco.2005.07.004

    work page doi:10.1016/j.parco.2005.07.004 2006

  37. [37]

    G. J. Percebois, A. Lacerda-Santos, B. Brun, B. Hackens, X. Waintal and D. Wein- mann, Reconstructing the potential configuration in a high-mobility semiconduc- tor heterostructure with scanning gate microscopy , SciPost Phys. 15, 242 (2023), doi:10.21468/SciPostPhys.15.6.242. 21

    work page doi:10.21468/scipostphys.15.6.242 2023